Pompeii/Amalfi in a day?

The main problem I foresee is the infrequency of trains. the Circumvesuviana runs once or twice an hour. Miss one and wait a long time for the next. I didn't take the buses, but they may also be fairly infrequent. Also, take into account that it's at least an hour by train between Naples and Sorrento. Also the trains don't run all night. You will have to take the last train from Sorrento in the evening to Naples. Will you have luggage? Where are you planning to leave it? If this is a high priority for you, I suggest you look into hiring a car and driver. Then it should be doable.

The Circumvesuviana rins every 30 minutes to Sorrento (you'll need to stop at Pompei Scavi, you can check luggage there). Then continue on the Circumvesuviana to Sorrento, where you can take the bus to Amalfi (a little more than an hour on the bus). So far two hours and a few minutes of travel time, three hours if you have to wait for connections. Then reverse the trip and double the time. The bus from Sorrento to Amalfi starts at 8am and does not run at night. I don't know where you can check luggage in Sorrento or Amalfi. You could check it at the train station or drop it at your hotel in Naples, but make sure you can retrieve it at the train station the night before your flight or make sure the luggage storage is open all night. Depending on how much time you spend at Pompeii and Amalfi, this could easily be a 12+ hour day. Chani's advice is the best, in my opinion. Also, be sure to arrange for transportation to the airport in advance, perhaps have your hotel reserve a taxi for you - I don't think public transportation will be running to the airport in time for your flight.

Unless you hire a driver, I wouldn't chance it, given that you have a flight to catch. Even if you didn't have a flight, you may not enjoy it. I saw Sorrento and Positano as a daytrip from Naples, and was not happy. These are places to settle into and relax in, not to whizz through. In your situation, I'd just see Pompeii and Naples.