Early arrival in Rome

If your flight is on time, it will take 60 minutes or more to clear passport control and get your luggage, then more time to get into the city. You might be able to be at the hotel by 10:00. Drop off your bags and let them know you have arrived. They may be able to make your room available right away or at least as soon as possible. By now you’ll want something to eat anyway, so if you cannot get into the room, go out for coffee and a pastry. Take a walk to get your bearings in the neighborhood. It is a very lively neighborhood and the Palazzo delle Esposizione is close by. They may have an exhibit of interest. Or just walk. It is the best way to experience Rome. By now, if you are not in your room yet, it is time for lunch anyway, and then your room will be ready!

We have always found that we could check in early so I would try to do that. If they say "No" then leave your luggage with them and walk the neighborhood. Might find an early lunch. The best thing for early jet lag is exercise and sunlight. You need the sunlight to help reset the internal clock. Personally we would check in at 2 and take a nap between 2 and 4. For past experience, our best practice for jet lag is the limited afternoon nap and this is after years of following the stay awake forever and going to bed early. We find the nap recharges our batteries just enough to get us through the evening and to a normal, local bedtime.

Hopefully the weather will be nice so you can stay outside. I don’t do major sites my first day, as jet lag does set in. Maybe walk to the Trevi Fountain, the Pantheon, sit at a sidewalk cafe and people watch, with a coffee. Coffee helps me.
I arrived in Rome at 7 AM once and had a miserable day due to jet lag. I’ve been back several more times with planned arrival for early to mid afternoon, no matter where I go in Europe. I can pretty much stay up until 9 PM when I arrive in the afternoon.
Others prefer morning arrival and don’t suffer much from jet lag.

As stated above, no need to try and hurry through arrival formalities, find you way to your hotel, drop the bags, check-in if you can (probably can't but absolutely you should let them know you're ready as soon as they can get to your room -- they should prioritize it).

Best ways to pass the time: being outside, being active, walking, getting fresh air and sunshine. Keep your body moving and your mind engaged.

Things to avoid: Sitting, being indoors, in quiet, dimly-lit places. Not the time to hit a museum or ponder the meaning of life in a church! Personally, although I find eating a light snack may temporarily distract your body from craving sleep, keep any foods/meals light - if you sit and stuff yourself, you're going to feel like it's after dinner on Thanksgiving: you may feel an overwhelming need to take a long nap (which you should resist).

Our flight to Rome arrived at 07:30 this past September.
We only had carryon luggage.
By the time we had deplaned, cleared Customs, found the right bus, taken the bus into Termini, walked to our accommodation nearby; it was about 11:00.
They let us leave our bags, and we walked down the street for coffee and breakfast.
Every hotel in Italy will let you leave your luggage till they can check you in!
One place a few years ago kept our bigger bags for us for 3 days while we did a side trip to Prague!