If you were to try and do this, I'd definitely go with a "fast train" train for Rome to Napoli Centrale and back - abt, 70 minutes each way - and a driver from Napoli Centrale to Pompeii and back. That would shave a chunk of time it would otherwise take if going the entire distance by road; far more time sitting in a vehicle than exploring the excavation.
You could also take the Circumvesuviana commuter train from Naples to Pompeii, as previously mentioned. The station for that train, Garibaldi, is connected to Centrale. It's a cheap, battered thing but gets the job done. Mind your valuables - as you should on ANY sort of public transit or tourist-magnet location - and you'd be fine.
But if you've not been to Rome before, there is plenty and more there to keep you occupied for 3.5 days without leaving the city at all. If you are planning to hit the most-visited attractions there (Vatican Museums/St Peter's. Colosseum/Palatine/Forum) you both may find yourself wanting some less strenuous activities during the rest of your time, as just those attractions require a lot of wear on one's feet. As already suggested, Ostia Antica would be a good choice if wishing to see a scavi as it's much closer to Rome than Pompeii and wouldn't eat as much time.
I guess I'm looking at the collective impact of your total plan versus sightseeing JUST Pompeii, if that makes sense?
If it's a must-do, then I definitely would hire a driver you could get
a round trip from Sorrento with a guide for about $200.
Um, our OP would be coming from Rome and not Sorrento? They'd have to backtrack to Pompeii to do it this way: time lost in what would be a long day to begin with.