In an old NYC apartment building built in the 50s. Would wires catch on fire if using a high powered AC?
Conduit may serve to contain heat in the event of extreme overload, but that's not its primary purpose (which is to protect the wires). Not all multi-family dwellings have wiring run in conduit even today. If your wiring overheats it's probably not protected with suitable breakers or fuses. 
Wiring and associate devices aren't rated for wattage, but voltage and current (amperage). Capacity depends mostly on the wire gauge and the circuit breaker rating. 110v * 20A = 2200 watts, but you can't always max out a circuit due to other usage on the same breaker. You may also have a 15A circuit, which would be overloaded (or nearly so) by your unit.


You should have a conversation with the building superintendent. Your concern goes beyond just this question. If wiring ever gets that hot, something's wrong.