Most of us here try to help as much as we can, but you're essentially asking us to google things for you. Break it down - you want a lounge. Check the AA site for any lounges at FCO based on that, then check the Amex Platinum lounge finder, then check the Priority Pass site.
Assuming you're flying DL ATL-FCO, you're likely to arrive at T3, based on what Flightstats is showing today.
You arrive into the sterile area and go through passport control, then get your luggage and go through customs. My last time through, I don't recall seeing an arrivals lounge - but even if there were, the lounges at FCO tend to be pretty bad, and I think all of them are post-security on the departures side.
There is a cab line right as you leave the terminal walking through customs. From there, it's a flat rate to any point within the city walls. That will honestly be more convenient than trying to Uber a car. You are far better off just heading to the hotel, cleaning up and getting a bite there or somewhere nearby.