How does musical notation in languages that use right-to-left writing direction (such as Arabic, Hebrew, Persian, Urdu, Yiddish) look like? Is it right-to-left too? If not (i.e. if they use the same European left-to-right notation), how do they manage to write the song lyrics under the staff to specify which syllable to sing on which note?
Yes, the standard international musical notation is used for songs in these languages as well. The words are broken up into syllables and written syllable for syllable below the staff.
The thing is that Persian is a language, just as math is a language and music is a language, so no matter what your daily language is, that language has no effect on math notations or musical notations.
For the matter of lyrics, well we manage adding the lyrics, like, above each "bar" (in music) we add it's own text. but it's not so important to be a concern.