If you arrive from the US with Alitalia or Delta, you’ll probably desembark from the departure level of the new pier (incoming passengers from the US don’t need to reclear security). If you’re flying AZ/DL business class, you are eligible to use Alitalia’s Piazza di Spagna lounge (if the lounge is not too crowded, they usually let arriving passengers in).
Otherwise, you can pay a Plaza Premium Lounge access (usually they let arriving passengers in), or use the HelloSky Lounge mentioned above (located in a public area, next to FCO’s train station).