Does reading "nasta'in" with an open "nūn" instead of a "ḍamma" in Al-Fatiḥah invalidate the prayer, and what must the worshiper do in such a case?

Your prayer is valid and you do not need to repeat it, because opening the "nun" in the word "nasta'in" (نستعين - we seek help) is a type of grammatical error (lahn) that does not alter the meaning.