According to information from the back of the Titanfall for Xbox 360 box, the game will require a 1GB installation for consoles that have a hard drive.
If you're able to snag a copy of Titanfall for Xbox 360 ahead of its release on April 8, you won't be able to play the game, as servers will not be turned on until next week. TItanfall is a multiplayer-only online game that requires you be connected to servers at all times.
The Xbox 360 version of Titanfall was developed by Bluepoint Games, not Respawn Entertainment, which handled the Xbox One and PC versions. The Xbox 360 version was originally expected to launch back in early March, but publisher Electronic Arts delayed the game twice to its new April 8 release date.