It is a 90 minute ride to the heart of Rome by coach but by train this is what I found:
There is a train station in the town of Civitavecchia with several trains per hour going to Rome. Local trains take you to Rome's central station (Roma Termini) in about an hour and 20 minutes. Faster trains can make it in under an hour. The station is close to the entrance of the port, but quite a distance from where passengers usually disembark.
Hope this helps!