Is that a green-red wire on the push button you want to replace? If so the push button may be wired to the normally open Coin switch shown on the schematic in the 120V AC section.
Palooka coin switch (resized).jpg
The 120V AC circuit can kill you so unplug the game from the wall before checking, measuring, modifying, working or futzing in any way with that switch or anything related to it.
With the game safely unplugged you can check if it's the same green-red wire tied to the Lock relay and Game relay latch switches with your multimeter. If it is the same wire, measure the resistance across the Coin relay coil solder lugs to get an idea what the resistance of the coil is. Using Ohm's Law (Voltage/Resistance = Current) divide the 120 volts of the circuit by the Coin relay coil resistance to get an idea of the current involved. Then double the voltage (to 250V) and triple the current for safety when checking the ratings of a new switch.
I'd also favor switches that have extra insulation or other ways of preventing the terminals from shorting to the coin door so that 120 volts can never come in contact with any of the door hardware.
/Mark