I bought Unreal Tournament GOTY, but I'm not sure how to install it with the native linux binaries. I downloaded the loki 436 goty installer, but I'm not sure what to do with it.
Any advice?

It probably isn't possible. The loki installer assumes you have the cds, which GOG doesn't have. You might try installing the game on a windows machine and then pointing the loki to that directory, but it's doubtful that'll work. The only other thing to do is use wine to play the game, which I've had some success with, though I haven't played with it recently.

You can do it, as I've posted here:
http://www.gog.com/en/forum/unreal_series/are_the_gog_unreal_tournament_games_linux_friendly
(Last two posts of the thraed)

Beg pardon, but that thread is for the 2k4 version. I keep getting this thread when googling for ut99.

I have followed this installation guide, seems to work fine:

http://www.pcgamingwiki.com/wiki/Unreal_Tournament#Linux

but don't try the Loki links on that page, instead do a google search for this filename: unreal.tournament_436-multilanguage.goty.run
Then follow the steps in the how to, and you get your GOTY version of UT99 working. It's how I did it, and spent the last 2 hours reliving a bit of my past and having a blast doing it. :)