No, you can not solve this without CB 2.x. The Facebook API has changed. There's nothing I can do about that. CB Connect was rewritten to use HybridAuth API library, which is a centralized API for multiple social sites and uses CB 2.x API. Please see our compatibility chart below.
www.joomlapolis.com/cb-20-addon-status
Please see the below video tutorial on what to expect when upgrading from CB 1.x to CB 2.x.
CB 2.x is completely free and your subscription already gives you access to latest CB Connect. So it costs you nothing to upgrade except for time to upgrade and double check everything is working as expected. As always be sure you backup your site before doing any major upgrade like installing CB 2.x.