To what extent is the hadith authentic that states: "Whoever is unable to fast, let him glorify Allah a hundred times each day to attain the reward and recompense of fasting the month of Rajab..."?

This hadith is a fabricated lie attributed to the Prophet, peace and blessings be upon him. It is part of a long hadith mentioned by Ibn Al-Shajari. The memorizers of hadith, like Al-Shawkani, have explicitly stated that no authentic hadith exists regarding the virtue of fasting Rajab, and that all narrations in this regard are fabricated or weak, having no basis. Therefore, it is not permissible to transmit this hadith except with a clarification that it is a lie attributed to the Prophet, peace and blessings be upon him.