Do I have to enable Remote Access to use Mobile Sync?

No. You don't need to enable Remote Access for your Plex Media Server in order to use Mobile Sync.

However, you do need to sign into your Plex account on the Server. The plex.tv servers store the configuration details for Mobile Sync and effectively coordinates the process between your Plex Media Server and your Plex apps.

Related Page: Sign in to Your Plex Account