Fiumicino is about 30 minutes from central Rome by train, so you do not have to sleep at the airport. Whether you should depends on the hour of your flight.
The train
The Leonardo Express is the direct service between Termini and the airport. Trenitalia lists about 32 minutes, EUR 14 per adult, and first departures from Termini at about 4:50am. Slower regional trains also serve the airport. We have not checked their times.
The taxi
Rome taxis charge a fixed fare between the airport and the centre. Ask the current figure before you book, and add traffic to the morning.
How to decide
If your flight leaves after about 10am, the train is a sound choice. If it leaves before 8am, you are relying on one of the first trains or a taxi. Read our early flight guide and the early-flight short list.