Is the circulated story about Eve true, that God punished her with menstruation every month after she ate from the fig tree, by saying: "By My Might: I shall make you bleed once every month"?

There is no authentic narration from the Prophet, peace and blessings be upon him, indicating that Eve was punished for eating from the tree. Rather, there is an account attributed to Ibn Abbas, may Allah be pleased with them both, that mentions Allah punishing her with menstruation because of it. However, this account is considered among the Isra'iliyyat (narrations from Jewish or Christian sources) which are neither to be believed nor disbelieved. Allah forgave Adam and Eve and accepted their repentance for their sin. As for menstruation, it is something Allah decreed for the daughters of Adam; it is part of their natural creation and not a punishment.