how to repair ATX PSU 5VSB problem
The good news is that the +5SB is universally a separate power supply. 
It is simple, but still sourced from the AC mains rectified, so as with all power supplies of this type it will bite you hard if touched. 

Here is a website with an introduction to a basic PC power supply, with a schematic. Your may vary but it will give you some ideas. 

If your power supply is good once powered up then most likely the fault is only in the SB power supply. 


Trace back your +5SB to find the regulator (almost always a linear regulator)
Find the unregulated supply for the regulator to see if it is varying or just the regulator output. 


There are not many components, but do be careful, the high voltage side can kill you.