Is the aforementioned hadith regarding the virtue of praying for the deceased on their first night authentic, and what is its legal ruling?

The hadith mentioned in the question is one of the fabricated and false hadiths attributed to the Messenger of Allah, peace and blessings be upon him. The prayer on behalf of the deceased in the manner mentioned has no basis, and it is not prescribed for a Muslim to pray on behalf of anyone. However, giving charity on behalf of a deceased Muslim is permissible at any time, and the Ahlus-Sunnah wal-Jama'ah (Sunnis) have unanimously agreed on the deceased Muslim benefiting from charity and supplication.