Instructions
Log your Xbox 360 into Xbox LIVE.
Download the NXE update if prompted by your console. If you aren't prompted, then you already have all of the necessary dashboard updates needed to download backward compatibility patches.
Load an original Xbox game into your console's disc tray and close it. The game will begin to boot up.
Highlight and select "Yes, update" when prompted to download the backward compatibility patch. The console will briefly download and install the necessary patch then restart. Afterward, you will be able to play the original Xbox game.