Instructions
Go to Xbox.com and see if the desired game is on the backward compatibility list. If not, you can't play the game on your Xbox 360.
Insert the original Xbox game into the console and turn it on.
Navigate to the dashboard instead of booting the game by pressing the Guide menu button on the controller and selecting "Dashboard."
Sign into Xbox LIVE by pressing the Guide menu button again, choosing "Profiles," then selecting your desired Xbox LIVE profile from the list.
Update your console by selecting "Update now" when your Xbox 360 automatically prompts you. If it does not automatically prompt you, then you already have the required system update to play original Xbox games. When the update finishes, your Xbox 360 will restart.
Boot up the original Xbox game by pressing the Guide menu button and choosing "Play" at the bottom of the list.
Select "Update now" to begin downloading the required update to play the original Xbox game. The "Update now" prompt automatically appears when the game starts. A progress bar will appear on screen and when the update installation is completed the game will reboot playable.