Frank, wha? ... ? ? There is no train in Positano. That's the whole reason this plan is un-doable.
hgerhard02, it would be helpful to have the month of the year, but you still won't get a Yes from us. I'm not in favor of the plan, but you could book a car and driver for 8 hours for about 250 Euros (ballpark estimate.) That's the best way to accomplish your exact plan, even though it will short-change both Positano and Pompeii. On the plus side, when you exhaust the same-old-international-retail-boutiques in Positano, you can find your driver playing cards with the other drivers, and get to Pompeii sooner than you expected to.
Edit: I can understand not wanting to take four people and their bags on the Circumvesuviana. We paid our hotel car service 90 euros to take us from Sorrento to the Naples rail station. But a major reason for sleeping in Sorrento is to have the best possible daytrip transit access. It's entirely practical to be at Pompeii one day when the turnstiles open, and do something else in the afternoon. And you avoid the worst of the sun and heat in Pompeii that way. I'm not telling you what to do, only suggesting that you seek some alternate solutions to your problem. The Circumvesuviana stops directly across the street from Pompeii's Porta Marina entrance (which has bag checking, BTW.)
I should also say that we paid for an 8 hour car and drive to see Positano, Amalfi, Ravello, and a bit more of the AC on a single day out of Sorrento, pickup/return from our hotel lobby. I'm not saying you must spend that much time to appreciate the AC, not at all. I'm trying to illustrate other ideas.