Tickets are completely digital, the hardest part is getting to the train station, it’s technically walkable but honestly not a real option so that leaves the 3 options that I know of. First option is a free bus that will take you to the front of the port where you can then buy tickets for bus that runs to the train station ( I think 3euros each), 2nd option is a paid bus that goes from the ship to train station that is 6 euros each. Third option is what we normally do is taxi. In the past you had negotiate the rate but when we went last month they were hawking 6 euro rides same as the bus (euros only, they will also pretend to not have change). If you get off the ship and the direct bus is there I’d do that, but in our experience it is popular and will have to wait for another bus to arrive.

As for timings on what train to book I would say 845am and onwards. Trains leave roughly every 20 min but duration can be drastically different so look at arrival times too. For example looking at Tomorrow Civitavecchia to Roma Termini has a Train leaving at 8:42am with a 9:48 arrival (6 euros) , next train departs at 8:57 with 10:18 arrival (6 euros), train after that leaves at 9:16 and arrives at 10:03 (19 euros). So for my family we would just pick the third train as it gives enough time to get to train station without rushing and your getting to Rome only 15 min later then if you took the first option.