Sorrento to Paestum in a day?

I can't say if your plan is doable theoretically, but it sounds nightmarish to me. My first question is whether you have checked the transit time lengths for all the legs of this plan. Second is have you checked the schedules to be sure you can get a bus/train/boat at a time that works with each part of your plan. Then, you'll know whether it's physically possible to carry out your plan. Not my style of travel.
BTW, you've omitted what in my opinion is the most beautiful spot on the Amalfi Coast, namely Ravello.

Kerri, your proposed day may or may not be physically feasible, if the transit schedules all meshed perfectly, but in any event I think you would be giving very little time to each of your stops, and spending a lot of time in transit and very possibly waiting in line for transit. July is high season, and you can easily find that you are standing in line for an already full bus in Sorrento, or Positano, then needing to wait for the next bus 30 minutes later. I agree that Ravello is a highlight of the Amalfi coastal area, and deserves a leisurely visit of at least a few hours. Certainly the Greek ruins at Paestum are wonderful, and the small museum at the site is excellent too. Will you be in Naples at all? I explored the Amalfi Coast while based in Sorrento, and took an easy day trip by train to Salerno and Paestum while based in Naples. Late in the day is the best time to visit Paestum, from the perspective of lighting, photography, and summer heat. Also, you should know that the Trenitalia and Circumvesuviana stations in Pompeii are some distance apart, requiring either a long walk or a taxi ride from one to the other. Paestum and Salerno are on the main train line, Sorrento on the Circumvesuviana. I hope you are able to do all this in one way or another, it's all beautiful and worthwhile.

Unfortunately, if you line up in July for the 8:30 Bus at 8:10, there is a very strong chance that you won't get on the 8:30 and will have to wait for the next one. If you are to have any chance of this very, very ambitious schedule, you should consider a 250 Euro car and driver. There is grave danger that he won't take you all the way to Salerno for an 8-hour booking, because it will take him so long to get back to Sorrento, empty. I don't think your itinerary is possible. You can search this newsboard for people who asked the same question before.

You have received excellent advice. The Salerno train, indeed, stops in modern Pompeii, not near the excavation, and I think, "on the other side" of it from the Circumvesuviana. Google Maps can help with this. We would have liked to do your plan, but even though we had five nights in Sorrento, we had to write off Paestum. Maybe on another trip.

We got good value for an Amalfi coast tour by car and driver, seeing Positano, Amalfi, and Ravello, with a free hour in each. Also a drive-by of two smaller towns, and back to Sorrento. That was over 8 hours, but he wrote it off. Pickup and return at hotel.

With a rental car my wife and I made the trip from Sorrento to Paestum via the Almalfi Coast. We got into Paestum late, had enough time for a quick look at the Greek ruins but we then stayed overnight and went back the next morning to get a better look and visit the museum. This was in May and traffic was a lot lighter than you will see in July and we felt rushed on the Amalfi Coast. In July, and with public transportation, you might make it to Paestum just in time to settle in for the night. That said, on a 30 day trip Paestum was one of my favorite stops in Italy.

This is Rick's note on the subject, which doesn't address how long you could spend at the connection points:

From Sorrento: While it’s technically possible to day-trip from Sorrento to Paestum by public transport (via Amalfi and Salerno), it makes for a very long day marred by worry about making connections back. Consider renting a car or hiring a taxi for the day. From Sorrento, Paestum is 60 miles and 3 hours via the coast (longer with summer traffic), but a smooth 2 hours by autostrada.

Just making a quick look at Circumvesuviana schedules and main line train schedules, it looks like the last return from Pompei Scavi to Sorrento departs about 20:15 and you'd have to leave Paestum at 18:00 to meet that. The 18:40 departure that gets to Pompei town at 20:12 (on my early June test date) won't work, because of the distance to Pompei Scavi station, mentioned earlier.