The story took place on an island, run by an avid hunter who
imprisoned his captives only to release them and hunt them down on the
island. The story is about a man who ends up killing the hunter, but
the twist at the end is that he had such an interesting time that he
simply takes the place of the hunter! It's a classic short story, but
I can't remember the title!

Hello.

I believe that you're thinking of "The Most Dangerous Game" (1924).  
http://www.classicshorts.com/stories/danger.html

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search strategy:
"the most dangerous game" "short story"

I hope this is the story that you had in mind. If not, please request
clarification. Thanks.

Wow, that was fast! You must really know your literature!

the story is "The Usurper" by Janvier Tisi.

Jason

I haven't read "The Usurper," but it would seem to be too new and
obscure to be considered "a classic short story."  Also, Amazon
suggests that "The Usurper" is set in the African country of Cameroon
(which is not an island).

The story is called "The Pest", starring John Leguizamo.  I read it
many times as a child.  A classic.