It looks like you are flying in from Florida, so jet lag may not be too bad. My hubby was nearly useless the first day, except he was able to walk around a bit that night and eat a bit of dinner.
If I were you, if you can fly in 3 days ahead then I would do that.
We flew in 3 days ahead and were able to do a Colosseum tour, Vatican tour, saw the Trevi Fountain and Pantheon. Late one day, we did a "Trastevere Twilight Food Tour" which was fabulous!
We walked to all of our destinations, except took a cab to get to the Vatican early one morning (it was dark), because we booked an early entrance tour. We walked back to or hotel.
We were there in October (2018) and the weather was very pleasant. We are from the northwest and I am afraid that June would be too warm for us. You would probably be fine since you are from Florida. We were in our late 60's when we did this cruise.