The B.I.R.G. is the Regional trains ("R" on the schedule on trenitalia's site) and includes buses/metro in Rome. A return ticket from Civitavecchia to the main station (Termini) in Rome, including use of buses/subways in Rome is €9. Compare that cost to what the cruise line want to charge you and you have your answer.
This entirely depends on if you are adventurous or not. We took the ship's tour & the guides were very interesting, something you don't get on your own, although there must be tourist leaflets in the city to guide you around the sights. the ship's tour will be expensive, though. Also there were a variety of tours on offer, all of which missed out some (differing) sights. we docked on a Sunday & was told that it was a reduced rail service to get to the city, which was quite a distance away, but I have read in these pages of people who say it is easy to do.
Providing you are not a lone traveller I would do it alone. Last year we were no the MSC Lirica and the Rome visit was quite expensive, but well worth seeing.
We went on a Sunday to Rome from Civitavecchia and it is very easy to do...you can find out the train times from the Trenitalia website...just get the shuttle to the port gate, turn right and walk along the seafront for about ten mins. to the station which will be on your left. Buy your tickets 9 euros return which includes your metro and validate it in the machine on the wall in the ticket hall. We went to Termini (which is the end of the line) and then got the Metro to the Colosseum which is directly across the road as you come out of the station...we then walked from there past the Forum to the Trevi Fountain (lots of sign posts directing you) after coffee and a cake walked to Spanish Steps and then got the Metro from there back to Termini . We got street maps of the internet...the Metro and train signs are excellent in the stations it was very easy to do (and we don't speak Italian!) A couple of stations before Terminii is S. Pietro which is what you want if you want to go to St Peters (as it was Sunday we decided not to do this.