You may run into some problems without the 'backwards compatibility' update, but I think the single player campaign will run fine. You may not be able to play the co-op even in split-screen without the patch.
You can actually create the update disc by going to the xbox.com website, then select support and search for 'backwards compatibility'. One of the links should lead you to a page with instructions on how to download the file and burn it to a CD which you can put in your 360. It will install all the 'bc' updates, including the CT ones.
I found the link to the page that tells you how to do it HERE.
