Considering the Hotel Relais Dei Papi in Rome?
A VT member wrote the following comment about visiting Rome:
early shuttle bus to FCO airport by jinwenwang
If the earliest train from central to airport (5:50am) can not catch your earliest flight (like mine), you can take the train from Tiburtina Station with earliest train starts at 5:05am to FCO airport; the second option is to take airport shuttle bus (blue) at 3:45am in front of Palazzo Masimo (left side of the Central Station) and it takes about 40mins to FCO airport.
I don't know the exact cost of the bus, I just gave the bus driver euro10 and ask him to keep the change.