There is a good mainline train service between Civitavecchia and Rome and when I was there on a previous cruise, many people made the trip themselves. Have a look at a Thos. Cook railway timetable - your travel agent should be able to help.
As an alternative, I can recommend nearby Tarquinia.
If you are familiar with Rome, or speak some Italian, or are comfortable about using those great European trains, go for it. You can really save.
Have a great tiem.