The T4 Arrivals Lounge (and T1 for that matter) are landside which means you have to pass through immigration and customs after your flight to reach them. You'd then have to pass back through security in Terminal 1 to catch your FCO flight.

Since you are unlikely to get in the T1 departures lounge for your FCO flight I'd get a shower and breakfast in the T4 Arrivals Lounge. If you are on the 292 you'll be able to get a shower right away, if it is the earlier flight you'll struggle to get a shower, breakfast and onto Terminal 1 for your FCO flight in time.