Is Metal Gear Solid 3 snake eater for Xbox compatible with Xbox 360

No.

Look at the official Microsoft website to see which Xbox Original games are backward compatible for the Xbox 360:
http://www.xbox.com/en-US/games/backwardcompatibilitygameslist.htm

Wikipedia also has an updated list:
http://en.wikipedia.org/wiki/List_of_Xbox_games_compatible_with_Xbox_360