No.
The Rome airport was a nightmare the last time we were there. Took forever to get through customs. Definitely not enough time 
We had a long layover overseas but barely made it to our connecting flight on time! I wouldn’t chance it.
What @Gourmetmom said! I would get to the gate and park at a cafe or browse the shops, which is what we typically do if we have a little extra time before our flight.
All this was predicated on the hope that we’d be arriving on time. But, lots to look into.
Last year, we were flying domestically and had a layover. After we got to our gate, DH decided to go get a cup of coffee. I told him we were supposed to board in a few minutes. He said he would be back in time. He asked me to watch his stuff and I said no!
Sure enough, we started boarding and he wasn’t back yet. I texted him to hurry. I got settled in to my seat, and next thing I knew, an off-duty flight attendant was sitting next to me! DH had gotten back to the gate just as they told this woman she could board. So DH had to take a later flight. Since then, he listens to me when I tell him not to chance it!
I am reading this: https://thepointsguy.com/2015/08/layover-guide-to-rome-fco/
Wonder how many hours they consider a “half day”?
Aperol Spritz, yes.
We’re the masters of short visits but I think to follow his ideas, you’d need a dedicated taxi driver.
There’s a lot of stuff in the terminal. I can’t promise you might not get bored, but it’s not the worst place to spend a few hours.
Thats my thought. Plus it looks like we have to take a shuttle to another terminal because there is a carrier change (they are partner airlines but the first leg is Alitalia and the second leg is a US carrier) so thats an extra thing to plan in. Plus, our first flight leaves for Rome at 5:15 am!! I am not thinking I’ll be looking to do much running around!
You don’t have nearly enough time to go to Rome. But let me just throw this out. Last time we were in Italy, for convenience I wanted to spend the last night at a hotel near the airport, and wife wanted a pool, so we booked 2 nights at QT Termeroma, 1 km from Roma Fiumicino Airport. Well, we get to this “hotel” and it is a luxurious Roman Spa and we didn’t leave the grounds for 2 days. Lots of pools scattered around the grounds, but an indoor area also. Many or the people there were Italian daytrippers, so you can go there for the day without staying in a hotel room. But I don’t think you have enough time even for this.
With a nice airport lounge and wifi, time passes enjoyably enough.
This is FCO we’re talking about; “nice” is relative.
There’s not; Flying Blue/Skyteam Elite Plus members use one of the Alitalia lounges subject to numerous restrictions.
The Plaza Premium Lounge, however, is quite nice, and has showers available, in addition to food/booze/wifi.
But bottom line, stay in the airport.
“This is FCO we’re talking about; “nice” is relative.”
Have you been lately? Much nicer than most American airports especially the food and shopping.
Yes.
Talk about damning with faint praise. 
Yes, FCO is better than it was, but still not near the level of other major European airports.
What are the restrictions to get into the Alitalia lounge @skieurope ? We are flying into FCO on Alitalia in business class, but then change to Delta (flying Delta One) to return to the US. So we will have just completed the Alitalia leg, but they are partner airlines. Looks like we have to take some bus shuttle thing over to the US airline terminals, but I am fine with hanging in the Alitalia lounge for a while if that’s doable.
I asked about your situation to the Delta Sky Club attendant in Seattle this morning. She said with Delta One/Business class on an international flight, you do have access to Delta Skyclubs and the Alitalia partner lounge. However, she said that even though that’s the way it should work, sometimes internationally they make their own choices.
So I took that to mean that they’re supposed to let you in for free, but depending upon who is working, there could be a “creative interpretation” that Delta isn’t responsible for.
I don’t have their policies memorized, but I believe access is complimentary flying in business class internationally with a Skyteam partner. Alitalia opened a new lounge this summer in Pier E, where DL flies from, and it is actually pretty nice. It’s definitely a pleasant way to kill a few hours.
If you have priority pass there are few lounges there, and as amex platinum holder you could also access their lounges too.