8 hours in Rome

Don't go to the Vatican museum after an 8 hour flight it will drain you as the crowds are bumper to bumper, even on a guided tour.
Put your bag in a locker at Termini train station. Rome is one of the great walking cities with stuff to see outdoors. Read the entry on wikitravel to see what you want to go see
https://wikitravel.org/en/Rome

Wow, this is kind of asking for trouble.
If you had been to Rome before I could see how it could work out well.

Never been before though and with no room to go to in order to freshen up a bit, relax at any point or lay your bags down ; going to feel more like a chore than a vacation I am afraid ; especially if you attempt the Vatican.
No matter how light you travel you will still be carrying more than a typical tourist without luggage will be.
Shame you didn't plan at least that first night there or just take an earlier train.

Given what you have stated I would do no more than walk around the main central area around the Pantheon, Trevi Fountain, Spanish Steps, etc... enjoying coffee at a piazza side table at least once.

Number one piece of advise is arrive with NO set firm plan like Vatican tickets. Impossible to know in advance how you will feel that morning.

I would never do the Vatican again! It was really packed, and I could not imagine doing it when exhausted I would have had a panic attack.

If I were you I would wander the city. Check at the Pantheon, walk by the Colosseum, and the Trevi Fountain. I think there might be a hop on hop off tour for Rome, that might be a good option for such a short time.

I think you have time for a nice introductory tour. Being outdoors is the best cure for jet lag, along with being interested and entertained.

Does your train leave from Roma Termini? You could go there by train from FCO and check your bags, then commence a tour. The problem will be reserving a tour in advance because if your flight is late, things can start to fall apart. And so many escorted tours start at 9:00 which would be hard to make with a 7:00 arrival and afternoon tours may not finish in time for you to pick up you luggage and make the train.

A private guide could be the solution. Perhaps arrange to meet a guide about 11:00, giving you time to get into Rome, check your luggage, and have something to eat. Engage the guide for 4 hours, then have a late lunch and retrieve your luggage and be on your way. Going rate is about Euro 60 per hour for a guide. LMK if you want a recommendation.

Thank you all for the suggestions. Although the Vatican museum sounds completely reasonable from the vantage point of my my pre-travel work desk, I have a feeling we'll be a lot happier with some unstructured, outdoor wandering (and a lot of espresso!) after 30+ hours of air travel (a mere 8-hour flight sounds like a dream come true!)