How old was Han Solo when Luke and Leia were born?

From Han Solo on Wookieepedia

Han was born on the planet Corellia during the waning years of the Galactic Republic around 32 BBY

From Luke Skywalker on Wookieepedia

The son of Jedi Knight Anakin Skywalker and Senator Padmé Amidala, Luke Skywalker was born along with his twin sister, Leia, in 19 BBY.

Which puts Han at about 13 years old when Luke and Leia were born.

Han was still on Corellia at this point, and as far as we know hadn't been off planet by then. None of the action around the prequels occurs on Corellia, so that's why we don't see Han in any of the prequel movies.

The canon Solo - A Star Wars Story - The Official Guide gives us a nice timeline of Han's life.

We don't have a precise birthdate for Han, but we can see from the chart that he was born 13 years before the Empire was declared (in 19BBY), the same year that Luke and Leia were born.

As to whether he appears in the prequel films, the answer is a firm 'no' although Millennium Falcon does cameo in Revenge of the Sith.

Han also makes an appearance in LEGO Star Wars: The Padawan Menace, a non-canon story set in the Prequel era.