If you're tired, you can also get a day room at the FCO Hilton. The rates are reasonable and it's nice to get a shower after a long overnight flight. Otherwise I wouldn't plan to spend time near FCO. It'd be better to have some time in Rome, and it's not like you can't have lunch around noon. Local lunch time may be closer to 1PM, but you'll have places starting to open between 12 and 1. That gives you time to eat and then get a cab back to the airport.