The apartment looks lovely and I love the area. We stayed near the Pantheon once and I made sure to go into it every day - such an amazing building. Great caffes and restaurants. I remember a good deli nearby - someone else will remember the name (it is very well known).
Don't be embarrassed about taking a tour! (A cruise maybe, but it is just me who will tut-tut.

) This is more like a day tour - completely different from 20 nights seeing all of Europe on a bus.
The short tour may be the easiest way to see Pompeii. Otherwise you would take the fast train from Rome to Naples, then change to the local train to Pompeii. If you were doing it yourself, you could overnight in Naples which is closer. We spent a couple of weeks in Sorrento years ago and liked it, but you aren't going to see much of it. If there was an option to overnight in Naples, I would take that instead, because then you could visit the archaeological museum which has a lot of stuff from Pompeii.
I wonder if you could do a massive day out and take an early train, have the day in Pompeii, then take an evening train back to Rome?
And, I am sure this has been suggested to you, you could go to Ostia Antica which is west of Rome and can be reached easily, instead of Pompeii. Not as interesting, but still pretty interesting. We were there last year and I am writing up a "Day Trip" for it now.