Hello,
Atari Side:
Blue=+12V
Red=+5
Black = Ground (COM)
on ATX side:
Yellow = +12V
Red = +5V
Black = Ground (COM)
You should buy an ATX extension cable and cut it. Cut Atari PSU Cable too. Then solder properly cables. Don't forget to use thermo-shrinking sheath. Even if the Micro PSU is well protected, it's better to test with a Hard Drive connected on it. Test Atari side pinout with a multimeter before connecting it to motherboard
And last, The green cable (PS_ON) on the Micro PSU must be connected to COM to power on. (with the hard drive, you can hear when powered on). Once everything is ok, you can use the Atari power button to connect PS_ON to COM.
Here is ATX PSU Pinout. I will try to upload a Photo of my Falcon later.