I can't help you in any of your question, but on the other hand I suggest you to take a trip on the following sites:
www.comune.napoli.it (city gateway)
www.gesac.it (international airport)
www.trenitalia.it (italian rail company)
IMHO, though I've never been to Naples by plane (it's only 130miles from Rome and I usually get a 1h30minute trip by train), I should rent a car to get from the airport straight to Amalfi/Sorrento since there is no evidence of bus-shuttle services from/to airport to main station or going directly to Costiera Amalfitana.
I've no clue of how would it cost a ride on cab to Naples -a 5 miles trip- (or to the Costiera as well).
Last train to Salerno, checked through trenitalia.it seems to leave at 11.00 pm, maybe there are a fewer going southwards and leavin a little bit later (Direction: Sapri/Battipaglia/Reggio di Calabria) but you should connect elsewhere, in Caserta...Landing in Rome would had offered you a wider range of solutions.
That's no easy at all.