Looking at the MPU board schematics, if you go with a CR2032 battery all you need to do is remove resistor R1 to the right of where the original battery was.
Resistor R1 is used to charge the NiCad battery from the games 5 volt power-supply. Without the R1 resistor, the battery will be disconnected from the charge circuit.
Diode D3 just above R1 on the MPU board is already in series with the battery as a blocking diode. You do not need to add any extra diode with the battery.
Personally I replace diode D3 with a 1N5817 which has a lower voltage drop than a 1N4148 but if you're uncomfortable doing this, then just leave D3 alone.
If you're going with a super capacitor, you want it to be 5.5V rated. The charge circuit will apply more than 3.6V to it.
