Question: I am trying to make an original character who is blind
and hunts with bow and arrow from a flying mount. 
(Obviously, no existing animal can fly while bearing the weight of a human,
so let’s assume that the mount is a mythical/fantastical species;
for instance, a dragon or a gryphon.)

What physical qualities (internal/external)
would enable and enhance the archer’s feasibility and hunting prowess?

Answer: 
Yeah, her/his other senses would need to be heightened, either by natural or unnatural means. Think Daredevil for instance. Other than just hearing, his heightened sense of touch allows him to feel variations in temperature/pressure/humidity, his heightened sense of smell/taste allow him to smell/taste particles in the air. These could potentially be factored into your characters story. 

Like I said, it could be natural through training and adaptation, or unnatural through magic or technology.

This is all assuming the character is human. If the character is another species/race such as one with larger ears or longer fingers it might bode well for his archery capabilities. And if the character is not human, you may consider giving the character a creative new organ. Think birds who can "see" magnetic fields which allow them to traverse the globe easier.
How about ability to see/detect minute alterations of Earth's magnetic field or electricity? That might allow him to detect live beigns from a distance. I assume a medieval background due to the presence of mythical creatures so no radiowaves everywhere to drive him crazy, though his perception of a thunderstorm would be interesting.

Can't imagine the kind of organ which would allow, I can only think it would have to be duplicated to allow triangulation for distances. Maybe replacing the eyes?