It seems like a lot in hot weather in a short amount of time. I would suggest no more than 3 nights in Rome that time of year. Depending on what type of traveler you are, how you respone to jet lag, I can tell you that last Sept. I arrived in Rome, then immediately took a high speed train ( about 1hr 15min) to Naples, stayed one night to see Archeological Museum and Herculaneum, then continued to Capri. I have to warn you, Pompeii is massive and extremely hot that time of year, also, with very little shade. You may want to consider Herculaneum, it can be seen in 2 hours, which leaves time for something else! It is also much better preserved than Pompeii. https://en.wikipedia.org/wiki/Herculaneum
Unless you are interested in hiking up it, Vesuvius can been seen from anywhere in the Naples area, so a visit may not be necessary. The archeological museum in Naples houses all the artifacts recovered from Pompeii and Herculaneum, so that might be a good way to cool off in the middle of the day if you are not at a beach. If going to beaches water shoes are a must as large rocks beneath the water make wadding in and out impossible.
Depending on whether you are an island person or would prefer to stay on the coast and town hop, you should choose EITHER Capri or the Amalfi Coast, for me Sorrento is very skipable compared to the other two.
I stayed 3 nights in Rome at the end of the trip, but was visiting relatives, so two nights may be enough for you. Also, relaxing first, and getting over the jet lag will enable you to be more on Rome time zone for a city visit, which requires LOTS of walking. Enjoy!