Disconnect old AT PSU first from mains plug, then you can safely disconnect the four wires going to the switch (blue,white, etc).
Then connect the ATX adapter green/black to any two that work, you cannot break it. Based on the video I guess the pins with blue and white that are side by side on the foreground, or the other two (black/brown? I can't see) on the background. Leave other two disconnected. If you have a multimeter, you can check which two are connected together when switch is on. The switch is supposed to connect ATX supply black (ground) and green (PSU_ON) wires together to turn the ATX supply on.
alexanrs wrote:I guess the old one has four because all the power from the PSU needs to run through the switch, and by dividing the current in two pairs of wires/contacts there is a lower risk of those wires getting overloaded.
No, it's to prevent you to get killed by electricity. Actually same current flows in both halves of the switch, but in opposite directions. In some parts of the world, your mains plug can be either way in the socket so which wire is live (has lethal AC voltage) and which wire is neutral (zero voltage) is unknown, so you have to switch both wires before it is safe. Compare this to common light switch in your living room which only switches one wire. The electrician has to make sure the switch cuts the live wire, so that when you turn the switch off, it is safe to change the light bulb. If the switch accidentally cuts the neutral wire instead, and you change the light bulb and accidentally touch the lethal AC voltage and something else that is grounded, you get a shock.