By the time we discovered cruising about 5 years ago, our two eldest were not holidaying with us although daughter has been 3 or 4 times with us. She absolutely loved Royal Caribbean (Voyager) and we believe it offers the best for young teens, which yours more or less is. NCL (Jade) looked well organised for young folks, as did P&O Ventura.
Both Barcelona ond Rome are fantastic cities, whichever you choose. There is more to do though in Barcelona that would suit an early teen than Rome. e.g. Barcelona FC Nou Camp stadium, atmosphere of Las Ramblas, the marina area and even the beaches which are very good. Also there is a theme park within a few miles of Barcelona, cannot remember the name though. Rome has none of that. The last time we were in Rome was October past and our 17 year old daughter was impressed, but even a couple of years ago at 15 she would not have enjoyed it.
Certainly there is something out there for you, just need to search and find it!