The trip from Rome to Sorrento can be long and complicated: take into account that it will take at least 4 hours from when your flight lands, if you are lucky with all your connections.
You cannot purchase train tickets in advance, as your flight may be delayed into Rome making you miss your connection at Rome Termini train station. Trains from Naples to Sorrento cannot be booked in advance.
From Rome Fiumicino, you have to take the shuttle train to Rome Termini, then the high-speed train to Naples, connect at Naples to the Circumvesuviana train that departs from the underground platforms, and proceed to Sorrento
The Circumvesuviana Naples - Sorrento line is always very crowded, often standing room only, and without luggage storage space or air conditioning. In addition, the Naples Piazza Garibaldi station is chaotic and pickpocketing is common.
Booking a private transfer to Sorrento is the only option if your flight lands in the late afternoon or evening, as the final trains to Sorrento depart at 9pm.
With public transportation: total cost of ticket is about Eur 60/person and the trip takes 4-5 hours.
With a private transfer including sedan or minivan with driver and luggage transport from the airport to your hotel, consider a cost of around €450 and a trip time of 3-4 hours.
If you were planning on visiting the ruins of Pompeii, consider booking a transfer than includes a stop at the archaeological site to make the most of your time!