I would go with MomCat's suggestion in her first paragraph - it's certainly doable in one day if you make an early start, but it would be a 'big' day. The difference I would suggest is that when you get to Sorrento you also have the option of getting a taxi to Positano if you want to avoid the SITA bus. Positano is a separate town to Sorrento, but the trip between the two is not too far. I'm not sure what the actual cost of the taxi would be, obviously it would be much more than the bus, but at the end of the long day you might decide it's worth it.
From Rome to Pompeii there are different types of trains that operate at different speeds. It's probably worth taking the fastest train to maximise the time you have left in the day.
When you get to Positano a taxi could take you much closer to your hotel than the SITA bus, which only has two drop-off locations on the 'main' road above town, and then you have to walk down to your hotel with your bags (unless you can catch a local taxi from the bus stop). If you catch the SITA bus, be sure to know in advance which stop to alight at, to be closest to your hotel. The two stops in Positano are called 'Chiesa Nuovo' and 'Sponda'.
The problem with visiting Pompeii on your way back to Naples is that you wouldn't have the chance for a shower & freshen up between walking around Pompeii and your flight.