PS_ON doesn't have to be grounded, it just needs the voltage on it pulling close to 0V. You can delay the pulse in many ways but before we explain how, we need to know a little more. When you say "after pressing a button" do you mean it should switch on after the delay and stay on, in which case how do you turn it off, or do you mean it should switch on after the delay but switch off again when the button is released?
Can the circuit use the 5V standby supply, which should be present all the time?
Brian.