The Port Authority of Civitavecchia provides a free shuttle bus for all cruise passengers. This bus takes you in a few minutes from your cruise ship to Fort Michelangelo at the entrance / exit of the port. From there it's only 200 meters on foot to the center or 1 kilometer to the train station.
In the beginning the shuttle buses run very frequent . Later, they usually wait until the bus is full. The name of the cruise ship is indicated on each bus, so you can't miss. You can return to your cruise ship by the same bus from Fort Michelangelo.