It is very realistic and sensible for you to consider spending one night. The distance from Rome is just soooo far.
There are 3 hour and 1:10 trains to Naples. But that still leaves you an hour from Sorrento, which isn't even "On" the Amalfi Coast. That's another 45 minutes (by car, usually, but a bus could possibly work in April) to Positano. I was underwhelmed by Positano, but it certainly was (like for you) a "must see" on our first trip to this area. BTW, that was our fifth or later trip to Italy. I don't consider it part of an initial visit to Italy, which is classically Rome-Florence-Venice.
I do not recommend staying in Pompeii. No one will ask you if you enjoyed your visit to Anaheim when you go to visit .... .... you know. Most people might stay in gritty, disorderly, historic, food-rich Naples, or Sorrento, where English tourists ask on Trip Advisor, "ow much is a Pint????" We spent 5 nights in Sorrento, because its so well located for daytrips. (It is NOT a serious "beach resort". And it's not very historic.) We did well, at a luxury cliff-front hotel, where we paid extra for a view of Vesuvius from our bed, and a balcony. (Ambasciatori Grande.) Either Naples or Sorrento provides a gritty, crowded, hot commuter rail line to all five of the most popular ancient "Scavi", including Pompeii and Herculaneum.
This is relatively expensive area. If you stay overnight, you may want to do all your visits on your own, to save money. Two tours, or even two 8-hour car-and-driver (the driver is not a certified guide) could easily be 400 Euros a day. A group tour would be less, of course. (We are independent travelers.)