Rome will be busy in August. Suggest you take a tour that includes St Peters, Sistine Chapel as they will have big lines to get in and with a tour you go to the front of the line.
A cheaper way you can take the train in to Rome from the port and they have a good on/off bus in Rome that takes in all of the sights ... but you could spend a lot of the day in a line which is a shame when you have limited time.
Cartagena is a great place. Suggest a tour to Murcia (half day) it's an interesting place then spend the day in Cartagena on your own. It's a very nice place and make sure you go up to the castle etc. With Celebrity they had a free shuttle between the ship and the town (not that far)which I think may have been provided by the town.
Livorno .. not sure as I will be at the port for the first time in November .. with NCL Gem and they have a tour that takes us to Florence with about 5 hours on our own to explore which I am going to do .. but I do know Florence a little so should be able to find my way around. I also know Pisa which is interesting but I think Florence has a lot more to see.
Sorry I have not sailed P & O so have no idea what tours they offer .. have a great tri.
Bearing in mind your comment about walking too far being a no no I would seriously consider giving Florence a miss. I was there in March and it was busy then. I have also been in August and the place was heaving with tourists (and it was very hot). Because of this, the tour buses have to drop passengers off about 15/20 minutes walk from the centre and (for a do it yourself tour) a guide takes you to a meeting point. There are electric buses that run around the city but are small and get full very quickly. We are fairly fit and walk quite a lot, but we still found it very tiring and had to pay extortionate prices for a drink. Pisa has the tower but once again gets very busy and has little else going for it (although you can get the train from Livorno to Pisa quite easily). We love the walled town of Lucca which is close to Pisa. Although you have to do some walking again, it is much smaller and the narrow streets give plenty of shade and make strolling around very relaxing.
We are going to Livorno in July on Island Star and shall get the train from Livorno to Pisa where we switch to the Lucca line. Each service runs at least every 30 minutes and is very cheap (warning validate your tickets before getting on the train by inserting them into the yellow box on the platform). If you decide to go to Florence take plenty of water with you and don't even think of queuing for the Ufizzi Gallery (even people with pre booked tickets had to queue for over an hour in March!). Otherwise, have a nice cruise.
Thanks for your answers,nice of some people to take the trouble as some answers are quite offensive,still some polite people out there, thanks again.