Train options are available and definitely the most budget friendly option.
To get from the train station to the port you can walk, take a taxi or use the special public transport line service that connects the Civitavecchia Train Station with the Cruise Ships and vice versa.
The service, which is exclusively for cruise passengers, operates from 7 a.m.-7:30 p.m. until 6:30 p.m.-7:00 p.m. on ship departure/arrival days only. The shuttle takes approximately 15 minutes to arrive at the destination and it leaves approximately every 20/25 minutes depending on the traffic and passenger pick-up/drop-off operations.
The shuttles leave from the small square in front of the exit of the Civitavecchia railway station.
The fee is €6 each way per person. Tickets can be purchased directly on board.
You can find all the info here. https://civitavecchia.portmobility.it/en/shuttles
Personally, I prefer to use private transportation to this particular cruise port. Certainly not the cheapest option, but I budget for the convenience of being dropped off at the port. For me, this is worth it and definitely less stressful.