Thanks for your input. Would the following schedule be more feasible?

Day 1: Arrive Rome in the morning
Day 2-5: Rome
Day 6: Train to Florence in the morning
Day 7: Florence
Day 8: Train from Florence to Rome in the evening
Day 9: Fly back home

How long does it take to get from the Rome train station to FCO (airport)? Would I be better off staying at a hotel near the airport if I have a flight at 12:30PM the next day?