Do you know of any good online resources that could be used to learn how to use wordpress to pull api calls from a rails app?
Don't even bother. You can find hundreds of people on elance to do it for you for a couple hundred bucks.
You'd want to do it as a WordPress plugin, this way you're not fucking around with any core files and it should be upgrade proof (for the most part). Start Here: http://codex.wordpress.org/Writing_a_Plugin
One small thing to take note of though with private plugins is to make sure the name of the plugin is unique enough that it won't collide with another one in the repository. A minor issue though.