How to diagnose and repair a radio?
gbarry suggests I post this as an answer. Thanks for the support, gbarry!

You can do quite some troubleshooting without complicated tools. For a start we may not even need the multimeter. 

If it works for a while and then stops working that sounds like some component is overheating. That would explain that you have to wait some time before it works again, so that it can cool down again. Try to find out if a certain part gets really hot. If you can't pinpoint a component that way, use cold spray to cool components selectively. Just spray on components until it works again. (You'll probably have to spray only briefly; the spray we use allegedly can cool down to -50 °C.) When you find the culprit it may be defective itself, or one of the parts surrounding it. Look for discolored resistors and leaking capacitors.