In theory you can do that, but...
... check what is the current requirement for the controller, because the 5V regulator in the arduino (NCP1117, datasheet here
) can provide only up to 1A. Consider the arduino itself can use up to 350mA, so that leaves only 650mA for the controller and anything else you have connected to the arduino.
12V of power input is the upper limit recommended for the onboard voltage regulator. If you feed it with 12V, it's good to check how hot it is getting from time to time, because it does have to dissipate all that power somehow.
If the controller has a current requirement higher than 650mA, my suggestion is to have a LM7805 (up to 1A) or LM317 (up to 1.5A) feed it instead, and use the 12V supply you have power the input of the LM7805/LM317 regulator.
Also make sure GND is common for everything: Arduino, 12V source, 7805/LM317 and PS2 controller.