Updating WordPress breaks custom theme

I have a client that I'm doing some work for. The only problem is his WordPress site is running WordPress 2.9.2. This is obviously an issue because of security, but also I'm unable to use the plugins I need to accomplish what he hired me for. When I update WordPress to the latest versions, it breaks his entire site because he had a custom theme created a while back.

Is there any way to fix this without completely redesigning the site?

Any advice or direction will be appreciated.