From Rome to Civitavecchia, you have several choices. First, there are seven train on a Sunday from Rome Termini (beginning at 6:09am) that will get you to the Civitavecchia train station by noon. Note that the train station is 5 long blocks to the port entrance. It's a bit of a walk. There should be a shuttle bus from the port entrance to the ship. The regionale (R) train costs 4.50Euro and the ICPlus train costs 8.50Euro. The regionale train is like a bus - no seat reservations. The train doesn't sell out. If no seats, you stand. It's a 1hr. 20min journey on the R train.
CLICK HERE to see what I see.
Second, you can take a bus from the Rome Tiburtina train station. Third, you can take a taxi.
On the return, it's a different story as there is most likely a pecking order as to who disembarks first (the higher-paying passengers first). If you get off early, you can make your way to the train station and get on the train. However, if you get off late, it could be problematic and you might have to share a limo service to Rome Termini or better yet, a limo service from the port directly to FCO. For international flights out, you will need to be there 3hrs early. The limo service is not cheap.
CLICK HERE for the limo website.