Another option is dlvr.it I use this to automatically push content posted on a website to Facebook and Twitter.
I think by default, you will have an RSS feed from your WordPress site. All you need to do is give dlvr.it the RSS feed URL and register your FB page, and dlvr.it will do the rest (it can also handle Twitter, if you want to do that as well).
This is a handy way to update multiple channels with the same content in one action. If you really want a random post, rather than all your content replicated, you would have to tweak your existing WordPress newsfeed, or set up another newsfeed that only includes random posts. I assume there is a WordPress plugin to help with that.