Wake-On-LAN can be done from an Arduino, and is a much more elegant solution to power-up. A similar “Shutdown” script could be triggered from the Arduino as well.
If you’re that intent on doing this through hardware, a simple logic-level MOSFET that shorts the appropriate pin on the power button should do what you’re asking. The switch is a momentary push-to-make type, but I’m not sure whether the ATX spec defines whether the switch connects a 5v pull-up to ground, or a 0v pull-down to +5v. I would imagine it’s up to the mobo which route to take.
On mine, the switch seems to ground a 5v pull-up. The proper solution would be to use an N-ch MOSFET to ground the lead that’s pulled up to 5v from the motherboard.
You can simulate this circuit with a 9v battery (assuming the role of the HIGH pin of the motherboard’s power switch connector) through a resistor (470R would be fine), to the anode of an LED. Cathode of the LED goes to the transistor drain pin. Source to ground. Apply +5v to the gate and the LED should light. This is your switch being “pressed”. Remove +5 from the gate and the LED should go out. (Releasing the switch.) Don’t forget a pull-down resistor from gate to source. (10k)
When it works (and you’ve worked out the pulse duration for on and off button presses), substitute the 9v battery / resistor / LED for the HIGH side of the power button. LOW side goes to the source pin, which should also be connected to the Arduino ground.
If your motherboard appears to use the reverse scenario (pull-down to ground that shorts to +5 when pressed), you’ll need a P-ch MOSFET with the gate pulled up to 5VSB, and a logic LOW to turn it on. The source pin will be connected to 5VSB (and your Arduino should be powered by that rail as well), and drain to the LOW side of the switch.
You can figure out which scheme your board uses by measuring the voltage of the power button pins (either) when the button is pressed. When pressed, it’ll be 0v with the N-ch scenario, and 5v with the P-ch scenario. You might use a 1k resistor between the switched side and your MOSFET until you’re confident you have it right.