Your ship will not dock in Rome but in the nearby city of Civitavecchia (yes, I know, almost impossible to spell and even worse to pronounce...
).
I don't think you would want to stay in Civitavecchia as I heard that it's nothing much to see there, so I recommend staying in Rome and then take some transportation from Rome to Civitavecchia (it's about 1-1,5 hours drive). I guess both train and bus will be good options but I have never done that myself.
From the railway station to the port in Civitavecchia you may need to get a taxi as the distance to walk is too long.
Maybe you can buy transportation from RCI? It will probably be expensive but will get you all the way to the ship. Don't know if they have that option in Rome/Civitavecchia though.
Transportation with a private car is about 120 Euros (about USD 180) for 2 people from a city hotel in Rome to the port in Civitavecchia. I found this on this website:
Click Airport Transfer: door to door private transfers airport taxi
Note that I don't know anything about the company, I just did a Google search and found them.
What ship will you be cruising?
/Erik