Of course it costs money to tie up a car and a driver while you sightsee, just as it does in the USA. Besides agressive Naples taxi drivers, you could ask your hotel who is their local preferred car service. That's what we did, and were able to put the car trips on our bill. If you finish in Positano, it might ? Even be cheaper to start with a local driver who is not stuck two hours from base when he drops you off!
You don't make clear if you have already decided against the gritty but cheap Circumvesuviana commuter train. You'll have to use an equally crowded bus from the end of the train line to Positano, which is only attractive [Edit: i.e. the bus is attractive only] for cost reasons. That bus often has a line in Sorrento longer than will fit on the next half-hourly bus. Positano is not well connected with Herculaneum, but I should also warn you that Herculaneum Scavi is a 15-minute downhill walk from the Circumvesuviana. While there is safe luggage storage at the Scavi, I would not make that walk with luggage. There is no food except vending machines inside the ancient site.