I cannot update the game right now, so would my xbox 360 version of the game play well enough, that I might patch it later, or do I need to get the latest patch before I begin any characters?

I would have to kindly ask my neighbor if I could bring my console over for an update, and that just feels weird.

Uh... I'd wait until you can patch it. Out of the box, you might get a gem (no bugs/few bugs) or a steaming turd (lots of game-breaking, crashing bugs).

Well, yeah. I started a character before any of the updates and can still play on the save.

You will have issues opening menus resulting in very low textures at times which will only be fixed by restarting the game entirely.