Civitavecchia Recommendations

Although it is not precisely on your question, I urge you to skim this long discussion, and then read what's important to you closely:

https://community.ricksteves.com/travel-forum/italy/rome-airport-to-port-outside-of-rome

I assume you are boarding a cruise ship. It is far easier to stay in Rome and catch a train to the port in the morning. Civit... is kind of a gritty port town with little to see or do. You could have a nice evening in Rome and a scenic train at the end to the port. The train between Rome and Civit... is basically a commuter run -- about three an hour - four at peak times and two during slower times. And the time range is 45 mins to hour 15 depending on the train and number of stops. The train station is close to the port and you probably will be able to see your cruise ship when you get off the train. You will not miss it since a lot of people make that run from Rome especially if more than one ship is in port.

You can catch a bus from the airport to Civitavecchia. Or you could catch a shuttle (RomeAirportShuttle.com) to Civitavecchia.

Another option would be to catch the Leonardo Express train into Rome, spend the night close to the Termini. A train to Civitavecchia is fast and easy--maybe 45 minutes.

Civitavecchia is a port city. They have a few hotels along the main road, but there is not too much to do there. As others have mentioned is to take the Leonardo Express from Leonardo da Vinci International Airport to Termini, get a hotel near the station and take the train the next day to Civitavecchia. At the Civitavecchia train station you can take a shuttle to the cruise port entrance for a couple of Euros. At the cruise port entrance you will take a port provided shuttle to your ship. Remember most ships don't start boarding until after lunch. You will have plenty of time to get from Rome to the port.