I've replaced quite a few PSU fans and you don't need to power the new one externally.
Cut the wires from the old fan right at the fan so you have some length to them. Then cut the plug off the new fan right at the plug.
Use wire strippers to remove the insulation from the wires about 1/8 to 1/4 inch.
Use automotive style "butt" connectors to splice the wires from the PSU to the new fan (you will need a crimping tool). Mount the fan and tuck the wires away from harm.