Finn has apparently had access to entertainment programmes (of a historical nature) as well as the more mundane First-Order weapons and tactical training.
As madness ebbed and surged around him, he wandered through the
village, feeling himself more a participant in a historical drama than
in an actual battle. The horrific and all too common red stains on the
ground contradicted his denial. This wasn’t like his training at all,
he told himself numbly. Unlike in simulations, reality bled.
Star Wars: The Force Awakens - Official Novelisation
What's most interesting is that in his personal study of "ancient history" he clearly views the heroes as being the Rebellion, not the Empire. An early (and evidently overlooked) seed of his own future rebellious nature.
“Huh?” If he had been bemused before, Finn was now thoroughly bewildered. Without thinking, he addressed the shaggy mass lumbering along in front of him. “Wasn’t he a war hero? In the fight against the Old Empire?”
Star Wars: The Force Awakens - Official Novelisation
There's also some indication that Han's activities may have been covered as part of Finn's overall combat training, especially since Han was involved in several iconic battles against the Empire.
For the better part of a week, they studied and were repeatedly tested
on different historical battles, many from the Clone Wars, some even
earlier.
Star Wars: Before the Awakening