JFK-FCO will arrive in Terminal C - You'll take a shuttle-train to main terminal. The London flight might not arrive in the same terminal.
I was in FCO last week and also flew back to JFK yesterday. The Lounge in terminal C is opposite C31. There are other lounges in the main terminal. Canarsie is right about the lounge - It is nothing to write about. YOu could ask them to meet you outside customs near the Poste (Post Office) in the Arrivals area maybe ?