In the past we humans thought that our planet was the center of the universe and everything revolved around us, due to science/math/astronomy and any other means we now believe that to not be true.
Would it be possible for a planet, Earth or otherwise, to be in the center of universe/galaxy/Milkyway/solar system and still have a sun? (the sun would need to revolve around the planet in the center)
Not as commonly thought of, no.
One of the things that we learn in orbital mechanics is that the planet orbits its star, not the other way around. More generally, the less massive body orbits the more massive body. When the difference in mass is large ($m_1 \gg m_2$), that's a good enough approximation, and for simplicity we can even consider the masses to be point masses; when the masses are of similar magnitude, it gets a little more complicated and the barycenter – the common center of mass – ends up somewhere between the centers of mass of the two bodies. When the situation is sufficiently extreme, even the formulas we take for granted break down entirely and completely different equations are needed to describe reality. Compare also the Newtonian and relativistic orbital velocity formulas (formulas 6 and 20, respectively) (Internet Archive link in case the page changes).
The only way to make a star orbit its planet is to make the planet significantly more massive than the star.
That, however, presents obvious problems, not the least of which is that by the time you go beyond 13 Jupiter masses, you get a brown dwarf instead. You would need a planet of, at the very least, very near that mass if you want it to be considered to orbit the star. What you would effectively have, then, is something very close to a system of two brown dwarf stars, one of which failed to gain the mass necessary to start its fusion processes.
On larger scales, it's not practical at all. There is no scientifically plausible way to explain why the Milky Way, with conservative estimates of its mass being 7e11 solar masses ($7 \times 10^{11} M_\odot $), would orbit the Earth, which measures about 3e-6 solar masses ($3 \times 10^{-6} M_\odot $). This difference is 17 orders of magnitude; said another way, given that the Earth has a mass of about 6e24 kg and a large satellite might have a mass of maybe 1e4 kg (10,000 kg), making the Milky Way orbit the Earth would be on a similar scale as making the Earth orbit a human-built satellite.
While in principle you can, as suggested by Separatrix, define the frame of reference such that the Earth is at the center, if you do so then you end up trying to explain relationships that eventually led to Kepler's laws of planetary motion and even later to modern orbital mechanics. Picking an Earth-centered frame of reference for your calculations will help with objects that are gravitiationally bound to Earth (which is why we often do it with Earth satellites etc.), you will have a much harder time using such a model to explain, for example, the movement of Saturn's moons. As a model will need to be a good fit for all available data, these headaches are likely to cause more grief than they are worth.
And of course, from a strict point of view, there is no such thing as "the center of the universe". For some elaboration on that, see for example What is in the center of the universe? on the Astronomy SE, or Does the universe have a center? on Physics SE, as well as the material linked from those questions and their answers.
You could conceivably have a long period binary star system, where both stars are roughly equal masses, with a "planet"* trapped at their gravitational barycenter.
This star system could reasonably even exist at or near the gravitational center of a dwarf galaxy, or globular cluster, or even a regular galaxy so long as you get rid of the super massive black hole that should normally be there. That could have happened as the result of a collision with another galaxy in the ancient past, which caused the black hole to be ejected from your galaxy.
As far as being at the center of the universe, if your universe is like ours, then everything in it is already at the center of its observable universe. And if your universe is infinite in size like ours is commonly believed to be, then it doesn't even make sense to think of it as having a center.
Of course, by a fairly technical reading of your question this answer doesn't work since the stars aren't orbiting the planet, but rather the planet just happens to be at the point around which both stars were already orbiting. The planet's presence likely makes no significant difference to them. This still works if you want the stars to have different masses. The planet just needs to be trapped in their L1 Lagrange point (but then the planet isn't in the exact (bary)center anymore).
*I say "planet" because this object probably does not satisfy the IAU definition of that word.
By using one of the fundamental principles of relativity you could argue that we are currently in fact the center of the universe. If our frame of reference is taken to be static, it does look like the sun is moving around the earth and all the planets follow extremely weird paths moving around the sun that is orbiting earth.
Describing a geocentric model like this physically however is not plausible because we know that that is not how the universe behaves. So while philosophically, the question does make sense and a geocentric model is feasible, it is unlikely to work on a physical level as explained in Michael Kjörling's answer.
In essence it is a question of simplicity and since the principle of Occam's razor tells us that easier explanations should be preferable if no extra depth of meaning is added through complications, the heliocentric model makes much more sense.
Following on from the straight up gravitational practicalities that Michael has described.
It's possible, but the maths is terrible.
You could cheat by keeping everything as it is, and using the arbitrary nature of co-ordinate systems to redefine the "centre" as being the Earth and calculate all the orbits relative to that. Redefining the datum to an arbitrary location is fine when modelling straight lines on an infinite plane, but an absolute pig when describing orbital motion. There is such a thing as a Geocentric Orrery, you wouldn't be the first to try it.
(If you really want to take this to the illogical conclusion, place yourself top dead centre on an immobile world and calculate relative to that)
The reason for making the Sun the centre of our solar system, apart from the fact that it is, is that it simplifies the calculations considerably. The same is true around the centre of the galaxy.
On a universal scale though, you're probably ok putting the Earth in the centre and plotting the relative movements of galaxies against it as we're back (as far as I'm aware) to linear rather than orbital movement.
One possibility not mentioned is that the "planet" could be an artificial construct, and actually be something more like a hollow Dyson Sphere with a giant star or even a black hole at the center; that way the gravity could be much larger than an orbiting body which would have enough mass to be a star, like a red dwarf (if the center of the Dyson sphere contained a supermassive black hole then the orbiting body could even be a normal-sized star similar to our Sun, and choosing a supermassive black hole would also avoid any problems with tidal forces on the surface of the Dyson sphere no matter how close it was to the black hole's event horizon, see this answer.).
As mentioned this FAQ Dyson spheres are not really stable since the net gravitational pull from whatever's inside the sphere cancels out, so if the sphere starts to drift relative to the object inside there's nothing to stop it from crashing into the object. So you would need thrusters of some sort to compensate for drift, perhaps venting some of the atmosphere surrounding the outer surface. The FAQ also mentions there are no known or theoretical materials that could form a rigid sphere at 1 AU (the distance of the Earth's orbit). Equation (7) on p. 11 of the paper "Dyson Spheres Around White Dwarfs" gives the needed compressive strength $S$ for a hollow sphere with density $\rho$ and radius $r$ around a central mass $M$, with $G$ being the Gravitational constant:
$S = GM\rho/2r$
We can use this to do some calculations:
For the upper limit of what might be possible with atomic materials, the paper "On the Strength of the Carbon Nanotube-Based Space Elevator Cable" discusses the theoretical maximum strength for carbon nanotubes as about 100 Gigapascals, or 1011 N/m2, which is the same figure given on p. 12 of "Dyson Spheres Around White Dwarfs" for the maximum strength achievable with matter held together by atomic bonds. "On the Strength of the Carbon Nanotube-Based Space Elevator Cable" also gives the density for this carbon nanotube material as about 1300 kg/m3.
If solve the above equation for $r$, giving $r = GM\rho / 2S$, and then plug in the above values for the strength and density of carbon nanotubes, we can find the minimum radius at which a Dyson sphere made of carbon nanotubes could exist around a central body with mass M, without breaking into pieces because the stress caused by gravity is too much for the material to withstand. I'll make this into an equation in plain text which you can plug into this online calculator and find the resulting radius for different values of M (which I'll express in terms of multiples of the Sun's mass, 1.989 x 1030 kg):
((6.67408 * (10^(-11))) * (M * 1.989 * (10^30)) * (1300)) / (2 * (10^11)) = 862858432800 * M
If we let M=1, the sphere would need a radius of about 863 billion meters, larger than the average radius of Earth's orbit which is about 150 billion meters. So, the Dyson sphere would need a radius of just slightly larger than the orbit of Jupiter (778 billion meters) in this case. And the higher the mass of the central body, the larger the sphere would have to be if it's constructed from carbon nanotubes. For example, if we used a supergiant star with a mass 100 times that of the Sun, the radius would have to be 86 trillion meters, about 0.009 light years.
We could also try plugging in the value for $r$ as a function of $M$ into the formula for gravitational acceleration, $GM/r^2$, to find the surface gravity on such a giant dyson sphere "planet" surrounding a central body.
((6.67408 * (10^(-11))) * (M * 1.989 * (10^30))) / (M^2 * 862858432800^2)
And here we find a problem: for M=1 (central body has the mass of the Sun), the surface gravity on such a sphere would be basically negligible, about 1.78 x 10-4 m/s2, compared to Earth's surface gravity of 9.8 m/s2. Even with the smallest object that might count as a star, which would have a mass of about 0.08 times that of the Sun according to this page, the surface gravity at the corresponding carbon nanotube dyson sphere would still be only about 0.0022 m/s2, still indistinguishable from zero gravity for any beings on the surface. I suppose you could build enclosed structures on the surface which would spin and create artificial gravity with the centrifugal force, but it wouldn't be very planet-like.
There is the theoretical possibility of matter held together not by electromagnetic bonds between atoms, but rather made up entirely of nucleons (like protons and neutrons) held together by the strong nuclear force, which at short distances has a much greater strength than the electromagnetic force. I discussed the possibility of objects made of this type of "strange matter" in this answer. Looking online, I found this paper which has some theoretical calculations for the strength and density of a hypothetical variant of this type of matter. On p. 508 the author calculates a strength of 7.5 x 1033 N/m2, and a density of 8.35 x 1017 kg/m3. So if we use these values and again find the minimum possible radius given a mass of M, we get:
((6.67408 * (10^(-11))) * (M * 1.989 * (10^30)) * (8.35 * (10^17))) / (2 * (7.5 * (10^33))) = 7389.6 * M
So if the mass were equal to that of the Sun (M = 1) this would imply the radius could be as little as 7389.6 meters (which is much smaller than the actual radius of the Sun, but you could also imagine a black hole with mass equal to the Sun at the center, with a black hole of that mass having a Schwarzschild radius of 2954 meters). And making the size larger than this will just decrease the required compressive strength, so this same material could be used to build a larger sphere around the same central mass, large enough so that the gravity at that point could be just 1 g. If you want to find the radius needed to have 1 g acceleration at the surface (standard gravity, or 9.80665 m/s2) you can solve the Newtonian equation $9.80665 = GM/r^2$ for $r$, giving $r = \sqrt{GM/9.80665}$, and again I'll put it into a form that can be plugged into that online calculator so you can play around with different values of M:
sqrt(((6.67408 * (10^(-11))) * (M * 1.989 * (10^30)))/9.80665)
For example, for M=1, the mass of the Sun, this tells you the sphere would need to be about 3.68 billion meters in radius. This is larger than the radius of the Sun (about 696 million meters), so in this case you wouldn't need a black hole.
If you want a very massive central body and a very large sphere, we can set the two equations above to be equal to each other and solve for M, to find a Dyson sphere where the smallest possible radius that can maintain structural integrity has a surface gravity of 1 g. It turns out that the central mass in this case would be a supermassive black hole with mass 247.89 billion times that of the Sun, and the radius would be 1.83 x 1015 meters (about 0.19 light years). So as long as the central mass is that size or smaller, it's possible to have a stable Dyson sphere made of nuclear matter which has 1 g surface gravity.
As other answers have explained, it is not reasonable to have a more massive object orbit a less massive one, and it would require nontrivial redefinitions of the coordinates.
However, it is not necessary for the sun to be the more massive object.
If we allow for astroengineering to create non-natural solar systems, it is very possible to make the sun less massive.
Stars are gravitationally powered fusion reactors, and the main reason why fusion reactors are so difficult to make is that the activation energy requirement of fusion is extremely high, requiring astronomical amounts of mass. However, if an artificial fusion reactor is used to power the sun, its mass can be far smaller (at the cost of a shorter lifetime).
Earth's sun burns 600 million tons of hydrogen per second. This means an artificial fusion reactor the mass of the Earth's moon ($7.3×10^{19}$ tons) will last for about 3500 years before running out.
I think the question was whether earth is the center of the universe, not our solar system (obviously not). The universe's background microwave radiation (Remanent from the Big Bang) is extremely uniform in all directions, as well as the Hubble constant (relative speed of regression of stellar objects). It sure looks like our galaxy is in the dead center. But the "Big Bang" expansion would include the expansion of spacetime, not just stuff thrown out into already existing space, rendering the question of "center" (point of detonation) moot or unintelligible because every point in spacetime itself originated from the initial point (look up "quantum entanglement"). So every point of space in the universe is the spacetime from the initial starting point of the universe. So, ... Wait for it ... Yes, earth is at the center of the universe. (but so is everything else!)
I have asked a similiar question on physics.se some months ago. (https://physics.stackexchange.com/q/246909/84895)
This is possible, and actually the case for a "otherwise".
For a earthlike and or planet this is simply not possible.
The required mass of your "earth" to be orbited by a brown dwarf, (And these guys aren't that bright at all) would be somewhere between starting fusion of a neutron star and collapsing into a black hole. And belive me. Neither a neutron star nor a black hole would support existens of anything your story could tell about.
It depends on how much you want the physics to be the same.
In a setting much like our universe, as others have mentioned, you would need a plausible explanation for a planet more massive than its star. Barring some kind of un-fusionable-tonuim in the universe, it doesn't seem like that's feasible. This is if we try to work with what we already know, and if we're not allowed to re-write science history.
If you have more liberty with how things work, or if there's more of a fantasy component to your setting, you could also regress to Aristotelian physics. It was also previously believed that there were mutable earthly elements and unchanging celestial elements and that they weren't made of the same things. You could establish that the Sun and the stars actually are fixed to a firmament which revolves around the Earth, and that the Moon, Sun, and stars are on three different paths, to account for the different orbital periods.
If that were the case in your universe, some of our history would need to be re-written. If the cornerstones of our Heliocentric model turned out to have the opposite results, such as the phases of Venus corresponding to the phases of the Moon, that could make for a very content-rich universe.
In short, it's possible if you either introduce some physics that we haven't discovered, or if you erase some facts that we already have.
Here's the idea on how it would work: An artificial sun, built by a sufficiently-advanced civilization. Said sun would be a satellite orbiting the planet, providing heat and light for one half at a time. For the sake of convenience, we will assume that the planet is Earthlike, and the satellite orbits it once every 24 hours, and provides a mean of 1360 W/m^2 of energy to the surface. For that, the orbital radius needs to be 42.24 thousand km. And to allow for variations in climate, we would need it to be a convex bulb, but a giant laser will work if you don't want frozen poles and an arid equator. And the giant laser would need to be able to fit the planet's radius within, which would be impractical af.
Using the mean, we can find that the artificial sun will need to provide a power of 1.73*10^17 W. That's nothing compared to actual stars, but it does the job. And a civilization advanced enough to build such a satellite would probably give it a power source. The dead center of the universe would be a perfect position to harness energy. Now, why would a civilization make a planet like that? Simple: It's easy to access, so they'd probably use it as a HQ for the influx of resources and citizens across their empire. And how they'd deduce that the center of the universe is where it is? They're extremely advanced, and have charted out the Big Bang.
And if it's where the Big Bang happened 13 billion 700 million years ago, maybe the planet is somehow containing and harnessing dark energy that's pushing for the expansion of everything. It could be like in the Tom Baker Doctor's final serial, "Logopolis", where the planet is hosting a mechanism that's basically a life support system for the cosmos. Or like in the Peter Davison story "Terminus", the planet is where the Big Bang happened because it's somehow connected to the creation of the universe. If the planet at the center of the universe is somehow connected to the creation and lifespan of the universe, it would be even more of an investment.
As the sun isn't natural, who's to say that the planet, or its position, is natural?
