The fall of '99 my husband and I stayed at the Pension Alimandi which is about 2 blocks from the Vatican Museum and very convenient to the Metro system. We arranged for the hotel to pick us up at the airport (I'm sure they would drop you off again). You can find more information in Rick Steve's Europe through the Backdoor Books. It was very clean. Enjoy Europe!
The airport for Rome is about an hour away from the heart of the very large city of Rome.
The Port is further from the city than is the airport.
If you were to go to the heart of Rome for the night from the airport, you would be traveling further from the ship rather than closer to it. There is a Sheraton Hotel which is quite near the airport. They call it as being in Rome but it is about 35 minutes from the heart of the city. (depending upon traffic) The hotel is a fine place to stay for one night if your purpose is merely to arrive at embarkation city a day in advance. It is full service hotel with restaurant etc and you probably could get a decent rate on line from one of the popular travel sites.
Assuming you are flying transatlantic, if you are only planning the one night there, you still could go into the heart of Rome for the afternoon/evening. And the next day when it is time to go to the ship, you would be closer to it.