The Infamy
The Kessel Run is obviously a major accomplishment that would make it known galaxy-wide. Just look to The Force Awakens. 30 years after The Return of the Jedi, Rey meets Han Solo and says:
This is the Millenium Falcon!? You're Han Solo!?
And then later,
This is the ship that made the Kessel Run in 14 parsecs!
Han corrects this to 12, but it goes to show that this feat was widely known since Jakku and Tatooine are not close together. Jakku is in the Western reaches of the Inner Rim and Tatooine is in the Outer Rim (see: Map of Galaxy).
Side Note: some comments have mentioned that a smuggler probably doesn't want their ship to be known. This example shows the Millenium Falcon is known by name only. Rey stole the ship from Unkar Plutt thinking it was a "garbage" ship and her only option for escape.
Legends Sources
The following was previously canon, but is no longer.
Han Solo wins his choice of any ship on Lando's lot in Rebel Dawn and picks the Millennium Falcon by name. Lando is described as being "dismayed" and Han is super excited to own it. Suggesting it has a reputation prior to Han owning the ship.
Han grinned, then threw both arms up into the air and whirled around in an impromptu dance, giddy with joy. Wait'll I tell Chewie! The Millennium Falcon is mine!
[...]
The Corellian felt physically tired, but so exhilarated that he knew he couldn't sleep - not yet. He had to savor his victory, his ownership of the Falcon, just a little bit longer.
Han makes modifications to the Falcon and begins taking other smuggling jobs.
Now that Han had the fast and (comparatively) reliable Millennium Falcon, he could take on the most challenging jobs. He still worked mostly for Jabba, who was basically running the Desilijic kajidic these days, but he also took jobs for other employers. The Corellian and his Wookiee sidekick became almost a legend on Nar Shaddaa as they broke speed records for the Kessel Run and flew rings around Imperial patrol vessels.
Note: this quote is indeed talking about speed records. The book describes Han racing his girlfriend, Salla, through the Kessel Run and beating her by fifteen minutes. It is only later in the novel that Han is desperate to escape Imperial vessels and goes "closer to the black hole clusters than any sane person would ever go."
Hey, Chewie. Look! He pointed at the instruments.
We set a record!
Chewie commented bitterly that their speed record had come at the expense of his nerves. Hans eyes narrowed. Hey, this is weird, he said. It says we actually shortened the distance we traveled, not just the time. Less than twelve parsecs!
Chewbacca thinks the instrumentation must have shorted and was off. Han stubbornly believes he did it in under 12 parsecs. The novel ends with Han landing the Falcon in Docking Bay 94 to meet with Jabba, where it is found at the start of A New Hope. Based on the Rebel Dawn events, the Millennium Falcon seems to have a reputation as a fast ship. However, the second part of Han's quote in A New Hope is likely him trying to make a name for himself and the ship, as he would have just recently made the record setting Kessel Run.
The proximity to the events of A New Hope do not match canon films like Solo, but Han's boasting is in line with canon sources. In Star Wars Adventures #15, Flight of the Falcon Part 2, Han, Leia, Luke, and Chewie land on a planet in search of a new location for the Rebel base (since Yavin's location was compromised). Han is eager to brag about the Falcon doing the Kessel Run in 12 parsecs, not even its involvement in the Battle of Yavin.
It is possible through sheer determination on Han's part that he has managed to makes his Kessel run infamous in the many years leading up to The Force Awakens.
True or Not?
It might be relevant to consider whether this feat was true or whether this is braggadocio, as comments have theorized.
Ultimate Star Wars states in Han Solo's timeline:
Kessel Run: Han Solo and Chewbacca fly the infamous smuggler's route in less than 12 parsecs, breaking a long-held distance record.
This would suggest that there was a known record prior to this and that Han was not the first to find a shorter route. It is a well known route, particularly among smugglers.
The truth to this claim does vary throughout canon. According to A New Hope: The Princess, the Scoundrel, and the Farm Boy, this claim is a lie which impresses young Luke Skywalker, but is obvious boasting to Obi-Wan Kenobi. However, both Return of the Jedi: Beware the Power of the Dark Side! and Solo: A Star Wars Story present it as approximately true. In the film, Han does the run in slightly more than 12 parsecs. So perhaps, Han is only slightly exaggerating that to less than 12 parsecs.
Comparison to Real World Ships
There are plenty of famed ships that are commonly known. The Titanic is very well known, even prior to its disaster, 100,000 people are said to have watched it depart from Southampton in 1911. Southampton's population in 1901 was only 104,824. (Encyclopædia Britannica) The Hindenburg disaster is similarly well known because at the time the first transatlantic passenger flight was a big story for the news (which resulted in lots of coverage of the explosion).
Outside of ill-fated ships, the SR-71 Blackbird is well known as the fastest jet in the world, even to non-pilots. Or look at the example you used in the question, "Lindbergh's solo flight across the Atlantic made the Spirit of St. Louis known worldwide." So it is reasonable to assume that the Kessel Run record would give the Millennium Falcon some notoriety. Particularly, as others have stated, around Mos Eisley.
TL;DR
The Millennium Falcon has a reputation as a fast ship, even prior to Han's ownership, and Han only improves that speed with his modifications. It isn't unreasonable for Han to think someone would have heard of his ship. However, people knowing about his Kessel Run might not be reality and Han is just trying to create that reputation for himself. Some sources even question whether he actually did the run in less than 12 parsecs.
