You really won’t need nor want a car in Naples itself. There are plenty of busses and metro to get you around the city if you prefer not to walk.
For the Amalfi Coast ferries and busses will give you all the options. The coast road is usually a traffic jam, you can only go as fast as the bus in front of you.
For Sicily a car will save time wandering place to place, but with only a week, pick either the Catania or the Palermo area.
This is the path I would take.
Napoli Centrale to Salerno by train. 42 minutes.
Salerno to the Amalfi coast villages by ferry, this will save time over the bus. Ferry is located about. 10 minute walk from the train station in Salerno.
https://www.travelmar.it/en/timetable-24-30-october. Travel timetable will change closer to your dates.
Spend 3 nights on the coast.
Amalfi to Capri by ferry. 50 minutes.
1-2 nites
Capri-Naples by ferry, 50 minutes.
1-2 nites.
Fly Naples to Catania 1hr. Easyjet R/T about $75
Or Naples-Palermo 1hr. Easyjet R/T about $70.
6 nites Sicily