I'm just posting this in the hope it may be of use to future travellers.
We booked our flights to Naples and hotel in Sorrento independently.
When we landed at Naples airport we got on the airport bus that was just parked outside Arrivals. The cost was €10 each and we paid the driver onboard. We caught the 11.00 am bus and it was practically full. I think the journey time to Sorrento was about 1 hour 40 minutes. (Out of this time we were stuck in traffic for a good 20 minutes going through a hillside coastal town - sorry - can't remember the name of it.)
When we arrived in Sorrento there was an 'Ask Me' information portacabin next to where we got off the bus at the town's Circumvesuviana railway station and we were able to pick up a map and get directions to our hotel from there.
For our return journey to the airport I was, initially, undecided whether to get the airport bus back to Naples airport - our flight was at 11.20 am - and, at that point of research, there were only buses shown as departing at 6.30 am & 8.30 am, or to get a taxi. I emailed a couple of taxi transfer companies that I had seen mentioned on here. Sorrento Cars quoted me €70; Johnnys Tour Service quoted me €80 and our hotel quoted us €90. (This is for 2 adults - hotel in central Sorrento - Naples airport.)
On doing some further research on here someone had put an update that there was now an airport bus departing at 7.30 am. Thanks to whoever did that post as that is what we decided to use as, on checking the bus schedule, we noted that it was due to arrive at Naples airport at 9.00 am which was perfect timing for our flight.
Due to the very heavy traffic near Naples (where the motorways merge) we actually arrived at Naples airport just after 9.40 am. (This was on Wednesday 18 July.) This wasn't a problem for us but I suppose the point I'm trying to make is if you do decide to use the airport bus (and for us it was a considerable saving) please, please, please do allow sufficient time. (There was a family on the the bus whose flight was at 10.30 am! That, to me, is cutting things a bit too fine! I'm not sure if they got the flight or not. Hopefully, they will have done.) If we had got a taxi we would have been stuck in the same traffic.
Just a thought but it may also be worth investigating how long it takes using the train to Naples and then the bus from Naples train station to the airport, especially in rush hour traffic.