3 hours is too tight at FCO. Between immigration (39 min average), the Leonardo Express (32 min each way), and security buffer, there's no meaningful time left in Rome.
Tight but possible from FCO with fast immigration. Leonardo Express to Termini in 32 min - a quick espresso and walk near Termini is feasible. Not worth it from CIA.
Enough for the Colosseum exterior, Roman Forum, or a walk through Trastevere. Leonardo Express each way is 32 min; Metro from Termini reaches most sights in 5-10 min.
Rome at its best: Vatican Museums, Colosseum, Campo de' Fiori, or Trastevere for lunch. Plan 32 min each way on the Leonardo Express and enjoy a full 6-7 hours in the city.