I remember when the original Unreal came out years ago I was able to run through the campaign with a buddy in co-op mode. My question is can you still play it co-op multiplayer?

Yes, it should work in the co-op mode just fine, but the expansion (Return to Na Pali) has some issues. Once you get to the intermission screen (where the player character is narrating), you can't seem to continue, since there's no exit from those small maps. I was able to finish the original campaign without problems though.