...I went to Sorrento (on my own!) in 2007. None of the hotels are that cheap, and they all seem to offer half board only.
I stayed here. Don't let the crappy music deter you. I chose it because it has a number of room options (I had a single, but I am sure there will be one to suit you), and it is very close to the train station (no, you can 't hear them. Sorrento is the end of the line) and to the centre of Sorrento itself.
Pompeii is just a few stops on the train, it has its own station (clever of the Romans to build it right by the station, I think). It is a marvel. One day isn't enough, I loved it....
There is also Herculaneum, another preserved site. In some ways better than Pompeii but much, much smaller. You can get there by train, too..
I think Rome by train is definitely do-able. You would have to take the train into Naples and change there, I think....