As I posted on another thread, I have had that start up/shut down shuffle twice.
Once m/b and once it was power supply.
You could try a cheap power supply and and see if it stays awake with that, which may eliminate one or the other.
If the ethernet connection is broken it sounds as though a new m/b is required anyway.
I suspect a capacitor is breaking down - can you see any bulging of capacitors on the m/b.
Just try with the bare minimum or you may overload it - even linux on a usb stick if you want a temporary operating system to test with.
Not sure whether previous poster is thinking about AMD APU's, the later ones sometimes need bios updates before they will run.
I don't think that is your problem.