WHen the button is pressed, the PSU should be on for a certain amount of time then shut off. Is this the correct way to do so?

 

Looks correct if you complete the ground connections on Q2 and the relay coil.
This also assumes the PS_ON to GND is correct to turn on the supply.

 

I would include a current limiting resistor in series with the Start Delay switch to avoid any damage to the 5V SB output.

 

You need a diode, cathode to +ve, across the relay coil to protect Q1 from back emf when the relay drops out.

Actually you don't need Q1 and the relay at all.... PS_ON is spec'd as a TTL level signal, with an internal pull-up to 5vSB. You only need to pull it to TTL logic 0 (<0.4v) to turn the PSU on. Just connect the collector of Q2 to PS_ON.

I'd put a 1k resistor between 5vSB and the switch.

The timing will be quite crude - how long were you expecting?

 

I don't know, the relay adds some crude hysteresis where using only the transistor may present some "unknown" voltage to the supply.

 

There is no 'unknown' voltage to be present. There's no magic about PS_On, it's a TTL input with a (typically) 50k pullup, designed to be driven by an open-collector output, which is exactly what we have here...

Here is the description of PS_on from the data sheet of a common control chip used in ATX psus:

"Remote On/Off logic input for CPU or controller. Turn on/off the PWM Output after the 7 msec
/ 25 msec delay.
PSon = 0 means that the main SMPS is operational.
PSon =1 means that the main SMPS is off and the latch is reset."

 

This configuration will give appx 10 minutes ON using a 470uf cap.