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.
Hidden power
=============

Your character has a hidden power that permanently allows them to see into the future. Hence their current eyesight has been traded for one out of sequence with time. They know where the arrow will be once they let go for they see it already having arrived. 

To make sure this is not OP this could be limited to an enhancement / curse given by the bow they wield and only bestows the future location of arrows potentially fired from it.