NEMA connector adapter
This should be on DYI, but I'll answer here anyway. First, assuming the griddle has no control electronics, just a heating element, connecting it to 120V would run it at 25% of normal wattage (half voltage into same resistance yields half amperage; since P=IE, wattage is one-quarter). I'm assuming this would not be acceptable.

Using a transformer to boost the voltage would likely overload the outlet and home circuit as it wold double the amperage draw at 120V.

There is one way to kludge (IMO) this to work. If your apartment is actually fed with two phases of 120V and you can find available outlets on each phase, you can build an adapter to combine the two out-of-phase hots to produce your 240V circuit. The simplest adapter, just wires, is dangerous as if only one of the 120V plugs is connected, you can have leathal voltage on the other. The safer way is to use relays so that the circuit only connects if both hots are plugged in. Search the web for "Quick 220" for more details.