Naples area for 3-4 days

• When are you going?
• Are you sure you want a car for this portion? Think carefully.
  
• Transportation is chaotic, confusing and can be frustrating anyway
  and with only 3 to 4 days I'd only stay in one place and do daytrips so you're not wasting half a day minimum on transferring to a new hotel.
• I'd advise Sorrento as a base as it is the most convenient hub for all the sites you want to see. We stayed in Sorrento for 2 weeks in 2023 and loved it. Plan carefully though, the transportation issues are real.

You have a lot of research to do, because this part of Italy has been starved of infrastructure funding for decades. It also tends to book up six months or more in advance. You also need to say where you are going next, since for most of this area, it is impossible to fly out of Rome the same day you wake up in Campania.

Maybe start here, just with the "drive Amalfi coast" keywords:

https://search.ricksteves.com/?button=&date_range=2y&filter=Travel+Forum&query=drive+amalfi+coast

You just need to ditch the driving idea (reasons delineated in other posts) and decide if the ferries would suffice for your needs.
When will you know if it’s 3 or 4? I’d be more inclined to split the stay with 4.
Who is we? Advice for a couple can differ from advice for a large group, etc.

You can stay in Naples and do day trips but instead of doing a day trip by driving to the Amalfi Coast, I would take the ferry to Positano or Amalfi town. It’s easier and the views from the water are great.

Another option is to split the nights between Naples and 1 or 2 nights in Amalfi Coast. If you do that, I would do Amalfi town. The bonus about this option is that by spending a night or two there, it gives you enough time in Ravello to visit Villa Rufolo & Villa Cimbrone, which are both gems with the postcard views the Amalfi Coast is known for.

Thanks all for the input— "we" are a couple. Husband's parents were stationed in Naples and lived on (in?) Possilipo and wants to show me and reminisce—open to other ways to get around and accomplish this.

Posillipo looks like part of Naples; I have no clue whether a car would be of any use to you there. Naples drivers are wild, and parking is a challenge in most large European cities. In any case, that makes three things you want to do in Naples. You'll pass through Naples on the train from Rome to wherever you opt to spend the night. I can't see any reason not to do what you want to do in Naples before moving on; I'd be looking to minimize time spent traveling back and forth on the train. If not for Posillipo, I'd recommend storing your luggage so you can go to the museum (which is large and requires multiple hours) and get your pizza on the day you pass through. With Posillipo in the mix, you might need one night in Naples.

I recall some earlier recommendations for the Chiaia part of Naples as a good place to stay. It's just east of Posillipo, so it might work well for you. Although I spent a week in Naples a few winters ago, I didn't get to either of those neighborhoods, so I have no personal advice.

Naples has a lot of worthwhile sights beyond the archaeological museum. I'll just mention one, whose location near the museum makes it a very convenient extra stop: The Capella Sansevero houses a stunning sculpture called "The Veiled Christ". The Capella sells out in advance (it was selling out about a week ahead of time in March 2023), so you'd need to get those tickets online before getting to Naples--and perhaps before leaving home.

The timing of your trip could alter people's advice.