Roma Pass: jury is out on that one but I wouldn't buy one myself as you still have to make advance, timed-entry reservations for the Colosseum and Galleria Borghese, and those seem to be the two attractions most commonly chosen by tourists using the pass for their two free entrances (72-hour pass). It doesn't cover anything at the Vatican, and please don't consider the Omnia Card as it's notoriously overpriced.
So, I'd just buy your reserved, timed entry tickets to the Colosseum (which includes the Forum and Palatine), Vatican Museums and Galleria Borghese in advance from their respective websites. Some of these tickets will disappear quickly once they become available so you'll need to stay on top of it.
Colosseum/Forum/Palatine:
https://www.coopculture.it/en/tickets/
Vatican Museums:
https://tickets.museivaticani.va/home
Galleria Borghese:
https://galleriaborghese.beniculturali.it/en/
Rome is very walkable so I sure wouldn't buy the pass for the transportation piece, which doesn't cover the airports. With some exceptions, your two feet will be the most enjoyable and efficient way to get around. :O)
What other attractions are you looking to see in Rome?
Editing to add one caution: June 4th is a first Sunday of the month when some of Rome's attractions will be free. This is NOT a day to go anywhere near the biggies, such as the Colosseum, and the Borghese still requires advance, timed-entry reservations for these days, even if free. As well, the Vatican Museums are closed on Sundays. This site lists some of the attractions which can be visited for free, although I'd check again closer to your travel dates in case those change:
https://www.romewise.com/rome-free-sunday.html
This might be a good day to do Ostia Antica: free on June 4,, not difficult/expensive to get to by rail, and I'm guessing won't be overrun. Shouldn't eat up more than 1/2 a day.
https://www.ostiaantica.beniculturali.it