Thanks Dan, I was about to modify my post and suggest priority pass
I was flying NYC-FCO-TLV on Alitalia, so flights landed and departed from Terminal 3 (non-Schengen flights).
The PP lounge there is the "Le Anfore Lounge", for which you have to take a shuttle to one of the satellite areas (if you're departing from the main building), and upon returning to the main terminal you need to go through security again...
The lounge itself is not exciting in the least - some baked goods, fruit and drinks. In my opinion, the chairs aren't even so comfortable. You also need to go down (and then up) quite a few stairs - no elevator or escalator available.
According to the PP website you should be able to access the other two lounges in the Schengen terminals, but it's a bit of a schlep from Terminal 3.