Is the Hadith that mentions the solar and lunar eclipses in the month of Ramadan as signs of the Mahdi's appearance authentic or weak?

The narration you referred to is not a prophetic Hadith. Rather, it was narrated by Al-Daraqutni on the authority of Muhammad ibn al-Hanafiyyah who said: "Indeed, for our Mahdi there are two signs that have not occurred since the creation of the heavens and the earth: the moon will be eclipsed on the first night of Ramadan, and the sun will be eclipsed in the middle of it. These have not occurred since Allah created the heavens and the earth." This narration is fabricated and falsely attributed to Muhammad ibn al-Hanafiyyah due to the presence of weak and abandoned narrators in its chain, such as Yunus ibn Bukair, Amr ibn Shimr, and Jabir al-Ju'fi, especially Amr ibn Shimr, who was described as a forger and a liar who fabricated Hadith for the Rafidah.