What is the proper protocol for applying to a postdoc sponsored by an one's undergraduate adviser?
I would suggest that you apply in the normal way, just as anyone else would. There may be rules for how that is done and you have to be attentive to them. Provide all of the required materials and make your best case for your application. 

But you can, in addition, send the professor a simple "heads up" email that you are interested in the position and are applying. Don't try to make your case in this email. Do it in the application materials. 

If you didn't have a previous relationship with the prof that you have maintained, such a letter would be presumptuous and ignored or treated as an irritant. But in the case at hand it would be a normal thing. 

You aren't taking advantage, exactly, since you have no control over the process.