The port which the ship leaves in is called Civitavecchia (pronounced Chi-tah-ve-kia) just in case you can't pronounce it and you get lost, you'll know how to pronounce it in Italian.
We decided to purchase the Celebrity transfer because we didn’t want to get have the hassle of trying to figure out train schedules, taking an expensive cab ride or even a private company’s transfer. From my research prior to the departing, it was my understanding the cheapest way is to take the train from Rome Airport (FCO) to Rome Termini Rail Station then to Civitavecchia (approximately a 40 minute drive from Rome) train station. From there you must find your way to the cruise ship from the train station. I hear there are shuttles from the train station to the ship from Civitavecchia terminal but I have not personally confirmed that. I hear you can walk to the station but given you have luggage good for a 13-night trip, it would be a hike with all that stuff with you. Therefore, I took the easy and a bit expensive route which is purchasing the ship board transfer for several reasons. 1) it’s guaranteed to get you to the ship on time. 2) When you get off the plane in FCO Rome, Celebrity cruise ship signs will be everywhere. 3) They will handle your luggage from the airport to your cabin without hassle especially after a grueling 9 hour flight.
I did purchase the optional panoramic tour of Rome with the transfer given that we were scheduled to arrive in Rome at 7:30am local time. I didn’t think going to Civitavecchia early and waiting there until boarding time of 2pm wasn’t ideal. I thought that maybe a good tour of Rome would give my wife and I time to see the city before boarding. Overall, I thought the panoramic tour was a bit of a waste. We did see the major sites (Colosseum, Old Roman Wall etc) but didn’t stop but only one place at Vatican City but only for about 45 minutes or so. It did not allow us enough time to purchase a ticket to see the site in detail nor the Vatican Museum (which I highly recommend to see). Also, taking photos on a moving bus wasn’t the best situation. Luckily, I have been to Rome a couple of times before. However, if you have not visited Rome before, I highly recommend doing a post cruise tour which I will go over in detail on Day 14 of this journal.