Using Zapier to auto post to your Facebook page
If you don't have already, create an account at Zapier. When you're ready, you can login to your Zapier account and then go to the following page to start syncing your RSS to Facebook page.
Click on the "Use this Zap". The rest of the process is rather simple and you can follow the instructions provided b Zapier.
Is Zapier free?
Zapier does have a free plan. So for a single site, it should be more than sufficient to sync your RSS feed to a Facebook page.