I've been trying to ignore this thread, but it won't go away.  It seems to
me that "a final arrow" implies that there are others that might be shot,
while "the final arrow" implies that it is the last arrow the archer has,
the last arrow in his quiver, the last arrow otherwise made definite.
 
You could make the shot arrow definite even if there were others left over,
but that would imply "He shot the final arrow that he shot," or some such
tautological nonsense.
 
So:
 
I shot a final arrow [and decided to call it quits].
 
I shot the final arrow [at my disposal].
 
I am ignoring such things as "I shot the final arrow of the tournament",
"the final arrow of the battle", "the final arrow of the hunt", etc.,
because these are clearly ordinally definite.
 
Tim