You can do it all, except for Venice. Make your way to Sorrento - consider a private transfer from FCO. Full price tickets from Rome to Naples are €48. The train from FCO to Termini is another €14. That's €120 and a train change to Naples, then shlep through the station to the Circumvesuviana to Sorrento, then get to your hotel, perhaps by taxi which is expensive in Sorrento. I don't have a clue how much a car and driver would be, but after a long-haul red-eye flight, it may well be worth the extra cost for a comfortable trip and give you a much earlier arrival in Sorrento. Plan to stay in the center of town, so you have easy access to the train and bus. A charming place with a view that's 1-2 kilometers out of town will be of no use to you.
Look at the weather reports when you get to Sorrento and consult with your hotel. You'll want to go to Capri on a good weather day - ferries won't/can't sail in bad weather. You will also want to take the bus to the Amalfi Coast on a good weather day. Parts of Pompeii are difficult for walking (these photos will give you an idea of why) and will be harder in the rain, but definitely doable.
Logistical details. Sorrento is built on the cliff sides. The center of town is pretty level, the train/bus is a 5-10 minute uphil walk, but the seashore - and ferry pier - is down a whole lot of steps. There are elevators in the center, but they may not be operating in off-season. You can download the SITA bus schedules from their website. There is one line that goes from Sorrento via Positano to Amalfi. If you want to go farther east on the coast, say to Minori and Maiori, you need to change buses in Amalfi. There is also a bus that goes up to Ravello from Amalfi. Trying to do all that in one day is really too much, but you do have a "spare" day for a second visit if the weather's good. Take the Circumvesuviana (CV) train to Pompeii.
If you don't have good weather, alternatives are to visit Herculaneum (I did it in the rain and it was fine) and/or Naples. The archaeology museum is a Wow and there are other sights as well. The RS book has a self-guided walking tour that begins across the street from the museum. In fact, you could visit the museum and probably do the tour on Thursday, if you take an early CV train and leave your bags in storage at the train station, then get a late afternoon train to Rome.
