Can suggest a way to get to Civitavecchia from Central Rome other than the train.? Please include cost.
If don't use the train, then it has to be a car service in the range of 125 to 150 E since it is about an hour from Rome. Try Romecab. Also see RoninRome.com for detail info getting there. The train is really you best option since the station right at the port.
It's much easier (but much more expensive) to use a car service. I can't remember the company we used, but the driver was even nice enough to take us to a supermarket to buy wine for our cruise on the way.
Except some cruise ship prohibit bring wine on board so you be sure you can bring wine before making the stop.