Learn Farsi: سایه‌ی او مستدام باد و بیمار مباد! - May his shadow endure, and may he not be sick!

bâd and mabâd are historical, defective verb forms used specifically for blessings, curses, and wishes (the optative mood). At this level, you already know to use the subjunctive (like bâshad) for wishes, but in fixed expressions or formal blessings like this, you will often see bâd (may it be) and its negative mabâd (may it not be). They are frozen in the 3rd-person singular form.

Does sâye-ye u mostadâm bâd literally mean "may his shadow endure"? How is it actually used?

Yes, literally it translates to "may his shadow be continuous." It is a very common, highly respectful Farsi idiom used to wish someone a long life and continued authority or presence. It implies that the person provides shelter or protection, like a shade tree, and you are wishing for that protective shade to last.

In the second clause, bimâr mabâd, there is no pronoun. How do we know who shouldn't be sick?

Farsi frequently drops pronouns when the context makes it clear. In the first half, the subject is sâye-ye u (his/her shadow). In the second half, the unstated subject shifts logically to u (he/she). Farsi speakers naturally understand that you are wishing for the person themselves, not their shadow, to avoid sickness.

I recognize the shape of the word mostadâm, but what is its root?

mostadâm comes from an Arabic root meaning continuous or enduring. You likely already know the related, much more common words (always/constant) or (continuation). Using here elevates the sentence, making it sound like a formal, traditional blessing.