Rome-sorrento-capri

Consider using this itinerary.

Day 1…
Arrive Napoli Central. Check luggage, using only an overnight bag or backpack.

Taxi to Molo Beverello port. Ferry to Capri 50 minutes.
2 ferries fit your timeline. 10:35 & 11:50

Visit Capri.

Ferry Capri to Amalfi 50 minutes.
17:50 & 18:00. Spend night in Amalfi.

Day 2…

Bus to explore Amalfi coast.

https://www.sorrentoinsider.com/en/sorrento-to-positano-bus-schedule?route=amalfi_positano_sorrento#section-schedule-result

Circumvesuvian timetable.

https://www.sorrentoinsider.com/en/naples-to-sorrento-train-schedule?route=sorrento_napoli#section-schedule-result

Can also be done in reverse, but may consume more time.

Edited to add

Based on what I saw when I was in Sorrento 2 weeks ago, I would seriously consider ferries instead of the bus to the Amalfi Coast. Buses were full and leaving people behind and I only imagine it getting worse. You could catch a ferry to and from Positano or Amalfi and then take the SITA bus between towns or there is a smaller boat company called Travelmar that works like a shuttle/bus between Positano, Amalfi and some of the smaller towns to the east. I just wouldn't trust the SITA 100% to get you back to Sorrento when you need to be.

To add to my other comment, I also was more impressed with Capri than the Amalfi Coast. Shorter trip, better views. My vote if you're pressed for time is more time on Capri and ignore the Amalfi Coast.

I somewhat agree with Allen, I recommend the bus because it runs later than the ferry

I, personally, would eliminate Sorrento, since it’s not part of the Amalfi coast, it’s part of the sorrentino peninsula.

I would rather use that time to travel by travelmar ferry all the way toward Salerno, visiting the villages, then train from Salerno directly to Roma Termini. This would give you a full day on the coast with no backtracking.

https://www.travelmar.it/en/timetable-from-april-2023

We are in the area now, and it's crowded in Amalfi and Positano. The lines at the bus stops are crazy long, so not sure that will be an efficient way to move between towns. As we continue in this rainy spell, folks were telling me this morning that the ferries to Capri were canceled, at least for the morning.

Mondo has a shared tour of the coastline that RS recommends, don't know if that is an option for you.
I had to be here to experience the full impact of the traffic, congestion, narrow roads, and overall difficulty of efficiently traveling in the area. We left at 8:30 am by private driver, spent one hr in Posiitano, one hour in Amalfi, and almost two hours in Ravello. Didn't return to our Sorrento hotel until around 5:00pm, so almost half the day in the car. It's a tiring day.
Because of the frequent rain, the experience wasn't quite what I had hoped. There were no blue skies and no blue waters . We are working on making lemonade out of all this bad weather.
However, We are enjoying just being in Sorrento.

I think your time schedule is unrealistic. Even the shortest ferry to Capri (from Sorrento) is almost an hour. And the island can't be completely seen (by land) in a single day. However, you can do it justice from 9AM to 5PM, or whenever the last three ferries are. (I wouldn't want to risk being on the "last" ferry of the day.

As noted, the bus line in Sorrento is always more people than will fit on the next half-hourly bus to Positano. And our host, Rick, has written that you should visit Positano first, because when the bus home (which originates in Amalfi) leaves Amalfi, it may be so packed to the rafters that it doesn't even TRY to stop in Positano. Get the message? Ferries are not always the answer, because (info from current travelers, please ... .... ) the dock repairs in Positano ran late this Spring, and that port can only take smaller ferry boats anyway.

What I'm saying that even with the step-child status of Public Transit in the USA, you cannot apply Los Angeles or New York transit rules to a service in an area of Italy that has been starved of Government funds for ... generations! You have made what sounds like a "perfectly reasonable", efficient use of two days. But it will be more like a continuous rush-hour transit trip at home, but for the entire day. You are also (I just mean, because you haven't been there before) under-appreciating how ... not "remote" ... but maybe, "limited access" is the actual Amalfi Coast. This is not like a daytrip to Lucca from Florence or something. It's more like a daytrip to Wright's Fallingwater, from State College, PA. (Dumb comparison. How about a daytrip to the Getty Villa from San Diego, which I briefly considered while on a business trip with my wife. I dropped the idea when all the 2-hour trips lengthened to 3-hours with actual Waze daytime calculations.

By the way, please put your home area into your visible Profile.

We spent five nights in Sorrento (because it's the best location for daytrips), and we got what we wanted, but it is in fact, a purpose-built postwar resort town, mainly reinforced concrete low-rise buildings. I'm not saying it's ugly. I'm saying it doesn't look like Band Of Brothers!

Tim, you explained it much better than I did, thx.