Does ATC handle non-regular widebody flights differently? Six weeks ago, roughly, a Corsair charter flight landed at Lamezia Terme International Airport (SUF), in Southern Italy. It was a Boeing 747-422, registration F-HSEA. It carried 500 tourists from Liege Airport (LGG), Belgium.

My question: when a flight like this is headed to an airport which is not an usual landing spot for wide-bodies, how does the ATC figure out a wide-body is coming? Do they figure this out by just communicating with the pilots when airborne or do they receive an official note beforehand?
No, ATC does not handle these flights differently from regularly scheduled ops
ATC isn't concerned with why an airplane wants to go to an airport or whether they regularly operate there or even if there are ground services available for them at the destination.
Whether the flight is regularly scheduled or not may fall under different regulations but ATC services do not vary.  The charter flight will be operated under instrument flight rules (IFR) and will have a flight plan filed with ATC prior to departure.  The controllers at the departure airport will receive information about the flight prior to that planes departure and will service the flight to its filed destination.  This works exactly the same way as normal scheduled service by that same (or any other) airline.