In Naples, you are in the city as soon as you leave the ship. You can either go on an organised tour to Pompei or Sorento. You can go to Capri independently, however due to the large number of tourists you might be better going in the organised tour.
In Nice, it is very easy to get to Monte Carlo and Monaco independently. If you do this you can decide where you spend your time. I personally would recommend Monaco. Hope this helps.
Rome...we found really easy to do on the train from Civitavecchia for 9 euros return including the metro and bus fares...Florence we did a tour where the ship's coach took us there and then we did our own thing before we all met up for our return. Naples we did the Pompeii excursion which was excellent. Athens I believe is easy to do from Piraeus but we did a whistle stop tour of the sites and then couple of hours shopping.
Naples is a bit like visiting the UK and going to Manchester. However one year we did an organised excusion walking to the top of Vesuvius, it's not to hard for anyone who can walk a couple of miles on the flat and the views were very good (just get the shoes a bit dusty so washable trainers or walking shoes are recommended). Sorrento is also only a short boat trip and far better just watch your timings if you go it alone.
As for Nice it depends where you actually dock/tender sometimes Nice is shown on the itinery and you actually tender at Ville France. Either way the whole coast is connected by a very cheap and regular train from Cannes to Monaco. We've done white water rafting from Nice, Marineland in Biot Antibes and the Grand Casino in Monaco all fantastic days and the latter two very cheap (in fact came back from the casino £150 up).