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I don't think the 5VSB is enough to safely power the arduino and a few small servos, so I think the most reliable way is to have an external power supply.
If you can open the PC's PSU for modifications, you can solder the AC input of the external PSU to the AC input of the PC's PSU (after the power switch, if there is one) so that you don't need 2 power cords. Then using a DC current sensor module on a 12V (for example) output of the PC's PSU so the arduino can detect when the PC is turned on or off