Holly - There is an earlier Circumvesuviana train to Naples. At Naples you could either get a taxi to the airport or the airport shuttle bus (Alibus) costs 3 Euro, pay the driver. Have not done the bus service so cannot give firsthand information, suggest you look at Tripadvisor for this.
There is a fixed charge for the taxi between the station and airport but you need to agree it before you get in the taxi. The taxi driver is supposed to show you a card listing the fixed fares. Good luck!
Site for the Circumvesuviana
http://www.vesuviana.it/web/en/homepage
Alternatively arrange a private transfer, it should cost about 80 Euros for two people from Sorrento to the airport. We have used De Martino, new cars and good drivers, but there are others who come recommended, again look at Tripadvisor. This is the solution we tend to use these days. A lot easier on the nerves!
http://www.autoservizidemartino.com/