Are you prepared to pay a few hundred Euros? If so, I'm sure it can be done. The big hotels in Sorrento have local travel partners like limo services - that way, they have a great incentive to make sure you are happy with the car service.
Is this your first trip to Europe? Your wife is not wrong, but most people can manage what you describe. Maybe driving in a strange place would be dangerous, but taking trains isn't that hard. It is true that the last leg, Naples to Sorrento, can be annoying, and not practical late at night. We spent a (pleasant) night in Naples because our flight from London came in so late.
It might be helpful, besides budget, to say how many nights you are in Sorrento, and how long your whole trip is. Will you spend any nights in Rome? Will you move to the Amalfi Coast after Sorrento? Are you using a Rick Steves guide book?
As a guideline, the Hotel Ambasciatori in Sorrento got us a car and driver for 8 hours on the Amalfi Coast for about 260 Euros. Since it's probably 4 hours each way Rome/Sorrento, maybe that's the ballpark. But you might be paying "Rome" prices if you use a service from there. In the scenario I've sketched, the travel service needs some protection from the possibility of your flight being cancelled or your not showing up for the pickup.