Amalfi Coast

I'd probably just drop the car as you arrive, but I don't think it's a terrible idea if you find a place with parking. However, it is really not that difficult to get around--you could store luggage at Pompeii. Otherwise, you will be leaving luggage in the car--something I would not do in Pompeii.
If you do it, I would stick to the mainland end--I know this place had free parking:
http://www.ilglicinebb.it/

In late April, the coast roads will be getting busier but won't be the mess they are May thru September. Stopping at Pompeii would be fine IF you remove your luggage and check it either at the Circumvesuvian station, luggage check is located downstairs in the tunnel under the tracks, or check at the entrance at of Pompeii.
Since parking will be closer to the station, I'd drop it here.

Thank you for your replies - I had not realized that leaving luggage in the trunk of a car at the Pompeii parking lot would be so risky. This leads me to my next question: where to return my rental car? With a cursory search I see three possible options 1) Drive out of the AC to Sorrento, drop car and then train to Pompeii, check luggage while visiting ruins and then continue on to Naples via train 2) Drive to Pompeii, check luggage during ruins visit and then drive to downtown Naples and return car 3) Drive to Pompeii, check luggage during Pompeii visit and then drive to NAP airport to return car and take a taxi to hotel in Naples for evening. I'm thinking dropping car in Sorrento and training to Naples might be the best option - what do you think?

There is a non-zero risk to leaving anything visible in a car anywhere in the world. This problem is not specific to Pompeii or even Italy. (AFAIK, there is no real "Pompeii Parking Lot". There may be some small multiple-place areas just off the street, but it's not like Disneyland. Most likely you'll be parking on the street, if you can find a nearby space.

I would not describe Positano as having good public transportation, especially in shoulder season, when some ferry routes with either be less frequent or not yet running. Sorrento has good public transportation, but it's not very much like Positano. And it's a good distance from Positano on a rather crowded public bus that runs, at most, twice an hour. Part of the reason the Amalfi Coast remains attractive is that it's a bit hard to get to and from.

If you got free parking, you could do Pompeii by car as a daytrip from Positano. In April, that's not a ridiculous idea, as it would be in May-September. You will have learned about ZTLs by then, so be sure about where you can drive the car. (We didn't have a car for our five nights in Sorrento, so I can't help on that.) Naples has a flat-rate system for airport taxis, so you want to download the rate sheet in case your taxi driver tries to add on some phony fees.

https://community.ricksteves.com/travel-forum/italy/how-do-i-get-from-pompeii-to-positano-is-there-a-direct-route

This is not the only or the best past discussion of some of your needs, but I suggest you look for some other discussions that may be helpful to you. The come up often here, and you can get a lot more by reading multiple threads.

This leads me to my next question: where to return my rental car? With
a cursory search I see three possible options 1) Drive out of the AC
to Sorrento, drop car and then train to Pompeii, check luggage while
visiting ruins and then continue on to Naples via train 2) Drive to
Pompeii, check luggage during ruins visit and then drive to downtown
Naples and return car 3) Drive to Pompeii, check luggage during
Pompeii visit and then drive to NAP airport to return car and take a
taxi to hotel in Naples for evening. I'm thinking dropping car in
Sorrento and training to Naples might be the best option - what do you
think?

Do not drop it in Naples city--risky traffic!
Check Salerno too, but #1 is probably best option.

Hello, I was just on the Amalfi Coast 3 weeks ago and loved it.
We flew into Naples and had a car service take us to the hotel. The streets were not packed.
We stayed at L'Ancora and it was absolutely beautiful and I think they have parking.
The rooms were spacious and clean and you looked right at the water below from a great balcony.
They have have a free breakfast buffet included in the room.
We walked out the front door of the hotel and a block up the road is the bus stop. The Sita Bus took us to the town of Amalfi, where we had a great lunch and shopped. The next day we took the bus up higher to the town of Ravello which was equally wonderful. So the bus system there is great.
You just have to stop at one of the shops to buy the ticket and wait by the sign on the road.
Hope you love it as I did.

Will you be in the area over April 25th (Liberation Day)? It's a national holiday in Italy, and I was visiting the Amalfi Coast April 26th and traffic was a disaster. If you can drop the car before going to Positano, I would do it (maybe Salerno?). The roads are tight even with limited traffic, and the best drivers in the world will still struggle because 1) they have never driven the road before and 2) there is plenty to gawk at out the window! Skip the stress and get a private transfer from Salerno or Sorrento to Positano and then utilize the SITA bus - it runs plenty in late April, or do a shared tour through Mondo (I did both of these). Ferries are running at this time, too, though a bit limited

Before you commit to Positano, read this thread. I'm not sure that traffic will be all that bad in late April - Easter's on the 12th so you'll be there after the holiday crush and before the summer surge starts in May. I'm partial to Salerno which is cheaper and more interesting than the AC towns with a good selection of restaurants and a long seaside promenade. The joy of the AC is the natural beauty, not the towns (which are definitely not among the most charming ones in Italy). That gives you the option to drive the coast if you are so inclined. You could also day trip by car to Pompeii and then return the car in Salerno and take the fast train to Naples. If you do return the car in Naples, try your best to avoid driving in the city (so far, the only city I've been to in Italy that I would never, ever drive in).