If you bought a later generation of the 360 it will already have the updates pre-installed. My Halo games played on my 360 right out of the box.
The games play exactly the same as they would on the Xbox. Graphics aren't jagged, they are the same as on the original. At least that's how it is for my 360 experience on these games (first two Halo games that is).
I know everyone has already said yes this game should work on the 360 and they are right.... kinda. This game works on any 360 bought before the most recent slim models. the slim models do not support the game or any updates for it, I found this out after spending 2 hours with microsoft trying to figure out why they wouldn't work when everyone said they would.