Positano to Pompeii

Best way: Stay in Sorrento or Naples instead
2nd best: Hire a car and driver for the day, hotel or car service
3rd Best: SITA bus to Sorrento, upstairs to Circumvesuviana train to Pompeii Scavi stop. Reverse for return. Expect crowds on the transit

Edit:
4th best: Use the safe, reliable bag check at Pompeii to see the site "on your way" to Positano. This has been discussed here in the past. It appears that a huge number of people do this. You have to think about whether you want to have a car pick you up after you're done in Pompeii, or pay him to wait while you see it. The issue is whether you want a cap on the time you're spending at a very large site.

5th best: Because it's always hard to meet a plane when staying in Positano, consider a last night in Naples or Rome, and see Pompeii, as just above, "on the way home." Many people fail to think enough about their departure-day schedule in beautiful Positano.

Time in Pompeii:
Most people spend two to four hours, but others post here that they spent the entire daylight day. In our case, we discovered that the rarely open Villa of the Mysteries was open. That adds (for walking reasons) at least an hour to a visit. Of course, you can skip anything. The question is whether you care about skipping, or whether you'll ever return to Pompeii.

Thank you, Tim, for the recommendations. We have our hearts set on staying in Positano so we will look into hiring a driver!

You can get a bus from Positano, get off at Sorrento, the last stop, then trains to Pompeii. Total trip about hour and a half. Be sure to get off the at the correct Pompeii stop, I think it is Pompeii Scavi, but can't remember. You will be only yards away from the entrance. On the way back be sure to get off at the second Positano stop or you will have a long walk down or need to take the local bus into Positano. Enjoy.

I took the train from Sorrento and it was not difficult - at the time it was not overly crowded (I went in Sept as well)...in case the hired driver doesn't work, don't hesitate with Option 3. It really isn't that big of a deal. "Pompeii Scavi" is the right station, not "Pompeii" the town.

You could probably save money by renting a car and driving yourself vs. hiring a car and driver for the day.
http://www.rentacarpositano.com/-en-Rent_a_car-2

You may want to check with your hotel in Positano, they may be able to arrange a car rental to be dropped off and picked up after at the hotel, I recall this being an available service through the hotels.
Some hotels have their own car available for rent, we stayed at the Hotel Eden Roc and guest could rent a tiny Twizzy car which was a 2 seater but tiny and electric. Not sure if had the range to reach Pompeii and back but they offered it as a rental to guests by the hour or day.

They also may be able to arrange a group tour with some companies as it is a quite popular location and time of year.

Sita bus from Positano to Sorrento 55 minutes 5 Euro per person.
Circumvensia bus from Sorrento to Pompeii 1 hour & Euro per person.
Taxi around 200 Euro.