Usually remote startup/shutdown doesn't require any support from the power supply. The remote controller and motherboard will handle all things needed. Even in a "normal" PC the power supply will only supply power when the motherboard so asks and nothing more.
The connector seems to have the normal red, yellow and black wires on it. That would indicate that it can be connected to regular PSU simply by connecting the same color wires to correct positions. Red is +5V, yellow +12V and black of course is the ground.
So based on the power supply being a regular desktop mATX power supply I would say it is highly likely you can switch it with another one and just wire the connector to the appropriate wires in the new PSU.