We spent 3 nights in Naples, but wish we had been in Sorrento. Naples is not a pretty city and HCBW really didn't feel safe walking around after dark.

On the first day, we arrived on the train from Rome and had the afternoon to explore Naples.
The next day we did a Pompey and Mt. Vesuvius tour, which gives time to hike up to the crater.
The following day we did the Amalfi Coast tour (small 12 passenger van).
The next morning we hopped on a train to take us to Florence.

I would not recommend driving yourself to see the Amalfi Coast: Crazy, windy roads and equally crazy drivers, very limited parking in the towns, not really knowing where to stop and what to try to see, etc. The guide and his knowledge/commentary made a difference when seeing the Amalfi Coast.

You could drive yourself to Pompey and the Mt. Vesuvius parking lot no problem.

I'm not sure how you could possibly see Pompey, Vesuvius, and the Amalfi coast in a single day, especially if day-tripping from Rome.
Seems hard to do without spending at least one night in a hotel down in the area.