We didn't use the train in Cittavecchia, but our cruise on Carnival last spring did end there. The train station was much further than a block. It looked like you could walk it, but doing so with luggage I think would be quite a problem.
Where we docked I did see a couple of taxis waiting, but certainly not many.
Perhaps someone who did do the train will post more info.
Take the free shuttle service to the port entrance. From there go out to the street and go RIGHT. The train station is a few blocks down on the left side and it is certainly walkable with luggage. We did this earlier this month. Train fare to Rome is 4 - 4.5 euros depending on which station you go to.