Would like a bus or shuttle service. The train option won't work as we will have luggage to manage & the taxi services are very
pricey.
I sorry but that is your only choices. The train puts you with in a half mile of the port entrance. A bus (if available) would put you into downtown Civit which is some distance from the port. There are no scheduled shuttles to the port from Rome. Something like Romecabs or Romeshuttle would be available but it is in the range of 150E because Civit.. is a long ways away - over an hour drive or train. Use the train and reduce your luggage. It can be done. We do it all the time and have made that run five times, 4 times with luggage. Think about reducing your luggage. We get by with one backpack style carryon each and one extra rolling carry on. Three pieces totally and no extra bags such as day bags, purses, etc.
I would suggest going to cruisecritic.com and finding your cruise line. Click on Roll Call. Find your ship and then your sailing. Through this group we were able to last month join together and get a limo to take us to Civ. If you need more info, email me privately at schwalmie at aol dot com
Most cruise companies provide transfers. Check with you're agent or cruise company. This will cost a far site better than 150E
The cruise shuttle for two will probably exceed the taxi cost and then there is the problem of getting to the pick up point.
Do as Jane suggests. Romecabs picked us and two other couples up from three different Rome hotels in a roomy late-model van. The driver was English-speaking and good. He handled all the bags. We were dropped at the ship. Total charge was 180€ + a 10% tip which isn't required but he deserved it. That was a couple of years ago so it may be more now, but totally worth it. Next time I'll go on CruiseCritic board and try to set up the exact same thing.
I used this service to take my mother and I too and from the airport tihs year. I would expect they also do transfers to the cruise ship. Professional, on time, clean etc... well worth it IMHO http://www.romeshuttlelimousine.com/ I think OP is just going to have to bite the bullet and pay (or pack lighter)