You don't have access to the Registry or any similar Windows magic, neither in W8 nor in WP8. There's only one exception I know of: Windows.Graphics.Imaging.BitmapDecoder.GetDecoderInformationEnumerator()
tells you both about mimetypes and extensions, but only of image formats installed on the local machine.
You'll probably need to manually create a dictionary which maps from mimetype to extension. Be warned: this is a neverending task. E.g. the mimetype "audio/x-mpegurl" has two extensions: .m3u and .m3u8. At the same time many servers claim that .m3u files have the mimetype "audio/m3u". So you'll end up tweaking and updating this dictionary over and over ...