Mathematics shows that we can make a one to one to one correspondence [of the] natural numbers with [the] even numbers. This is not right If there is a last number.
There are infinite sets that look exactly like the set of natural numbers N apart from having some finitely more elements after all natural numbers, and there being last ones: they are precisely the 'successor ordinals' of N: N+1, N+2, etc., and their forms look like
N+3 := {0, 1, 2, ..., N, N+1, N+2}
for example, so there is in fact a last element, to wit, 'N+2', and we still can put it into bijective/one-to-one correspondence with N, simply taking
f: N+3 → N
f(N) := 0
f(N+1) := 1
f(N+2) := 2
f(0) := 3
f(1) := 4
and so on
But it's obvious that the fact that there is also odd numbers implies that the set of all numbers is greater than the set of even numbers.
Proper containment is indeed one sense under which N is 'greater' than the set of even natural numbers (let's call it) E: since containment ⊆ is a 'kind of' ordering for sets, it's reasonable to interpret E ⊊ N as such, but as you seem to know, this is not really the same thing as 'size comparison'
If we introduce the time to reach numbers, we reach a different result.
In current mainstream mathematics at least, there is no notion of 'time' related to or imposed upon the objects of inquiry: they just kinda sit there statically. Even if we speak of, say, 'a function varying with (respect to) time', it really still is a list of ordered pairs, nothing more, nothing less, so that the analogy
E.g. if one is walking along a road to pick up fruits. [...]
doesn't quite make sense
So can we say that the one to one correspondence in set theory talks about nothing in real world?
One-to-one bijective correspondences for finite sets does correspond to (in a sense, is) our ordinary use of finite numbers to speak of finite collections in the real world, but yeah, since there seems to be no actual infinite things in the real world, such a correspondence can not extend in this direction