We went from the Port to the Airport with no problems.
Lesson learned. Taxi from Airport wouldn't take us to the hotel only 6 miles away. They wanted the high fares and trips to the city only.
We had to wait 4-5 hours (off season) to get the free shuttle from airport to hotel. We advised our neighbor who was traveling with a group they would be better off getting a Van to meet them at Port, since they were staying in the same hotel we did.
If you are in good health and can carry all your luggage -- there is no problem taking the train into Rome or the Airport. So many flights leave early in the morning and you have to spend the night.
If we did it will lots of luggage any healthy person can.