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.