Hmmmm, I would suggest taking the cruise line's transfer on this one. The airport is in the opposite direction from Rome and Civitavecchia is at least an hour's drive from Rome. There is a hotel right at the airport where they usually have everyone congregate, so it is an easy process.
Barring that, the train is a reasonable solution if you are on a budget, but shlepping luggage on and off the train can be an issue. The train station in Civitavecchia is only a block and one-half from the entrance to the cruise terminal.
A taxi might be a reasonable solution, but there are a good number of tansfer companies that will also take you. If you do not have any port time in Rome, I might consider a 1/2 tour of Rome picking you up at your hotel and dropping you in Civitavecchia.
Hope that helps.