Mojang said they were going to stop updating the on Xbox 360 edition. Is there a way to take the update (download) and update it manually?
No, there is no way to update Minecraft beyond the version that Mojang last released on the affected console. Without Mojang creating new updates for those consoles there is no new update to begin with that you could apply to the game.
Mojang decided to drop supporting those consoles as they were only a small percentage among active players. Source