You have already received some excellent advice. I will reiterate HOW HOT Pompeii will likely be. It was a decade or more ago, in early April, and the sun/heat was already intense for our visit......I can only imagine how HOT August will be. Others have mentioned, not much shade in Pompeii.
I also will suggest (plan ahead) to hire an INCREDIBLE guide (your hotel may be able to advise) to take you thru Pompeii (guide would typically meet you at the entrance). In a couple of hours, there is no way you can begin to see it all. The guide we had (part of a Tauck trip, but a local guide they arranged) was incredible, as she efficiently showed us key examples of different things while also talking about what life was like and how "advanced of a society" Pompeii was at the time...fascinating, adding along the way that there is much more to see, but she wanted us to get some basic highlights. It was, indeed, fascinating...not only the ruins of the structures but how each was "used" and how the people lived (then died). My husband commented he would like to someday go back and spend an entire day there (me, nada...the highlights were enough). Sun hats/bottled water/sunscreen, even an umbrella for sun protection. Cannot comment re: Mt.Vesuvius...we only saw it from afar.
Re: private driver. When we have traveled on our own (not part of a tour), we have found (for certain locations) private drivers can really made a HUGE difference in efficiency and comfort level. Re: safety of your belongings: If you book your driver thru your hotel, it would be highly unlikely you would need to worry about the driver getting into or making off with your luggage (I still would have the bags locked, though, as I think that is good practice in general). Do not worry about the driver getting robbed. It will be busy there, and fellow drivers typically know each other and watch out for each other, too.
Re: when to go to Pompeii.......directly from Naples or after settling in Positano.....just do it FIRST THING IN THE MORNING, when it is cooler, unless, of course, there is a thunder storm or other weird weather going on during your stay.
I would also suggest taking a (I can't recall what they are called), but a boat along the Amalfi coast to see the coastal communities from the water, too (entirely different, lovely views). Lunch in Ravello would be a nice treat. The views from Ravello (on land) are the "travel photo" one "always sees." What I do not remember is if the boat that goes along the coast could dock in Ravello or if we were transported from elsewhere. I loved the short visit we had there. We stayed in at a quiet hotel with incredible views right outside Positano (each room had a lovely large terrace), and the one time we went into Positano, it was amazingly crowded. I also recall traffic on the main road in from Naples was very heavy. If the hotel recommends a driver, and you can talk with the person ahead of your trip, he/she may be able to also advise re: what might be most efficient....and the likely the hotel can provide the same advice. I often ask from multiple sources (as you are with the RSForum) and see if I get the same answers each time. Often it is not just the answer, but the "why" behind the answer that is also helpful.
Enjoy your trip and do what is right for you. Safe travels!