Praise be to Allah.
There is no saheeh hadeeth from the Prophet that speaks of the virtues of the month of Rajab. Please see the answers to questions no. 75394 and 171509
Ibn ‘Uthaymeen (may Allah have mercy on him) said:
There is no saheeh hadeeth that speaks of the virtues of Rajab. The month of Rajab is no different from the month of Jumaada al-Aakhirah that comes before it, except that it is one of the sacred months only. Otherwise there is no fast that is prescribed in this month, and no prayer or ‘umrah or anything else that is prescribed. It is like all other months. End quote.
Liqa’ al-Baab al-Maftooh (174/26).
It was narrated by ‘Abdullah, the son of Imam Ahmad, in Zawaa’id al-Musnad (2346), at-Tabaraani in al-Awsat (3939), al-Bayhaqi in ash-Shu‘ab (3534) and Abu Nu‘aym in al-Hilyah (6/269) via Zaa’idah ibn Abi ar-Ruqaad who said: Ziyaad al-Numayri told us, from Anas ibn Maalik who said: When Rajab began, the Messenger of Allah (blessing and peace of Allah be upon him) said: “O Allah, bless us in Rajab and Sha‘baan, and let us reach Ramadan.”
This isnaad is da‘eef. Ziyaad al-Numayri is da‘eef and was classed as such by Ibn Ma‘een. Abu Haatim said: He cannot be quoted as evidence. Ibn Hibbaan mentioned him in ad-Du‘afa’ and said: It is not permissible to quote him as evidence.
Mizaan al-I‘tidaal (2/91)
Zaa’idah ibn Abi ar-Ruqaad is more da‘eef than him. Abu Haatim said: He narrated from Ziyaad al-Numayri, from Anas, odd marfoo‘ hadeeths, which we know only from him or from Ziyaad. Al-Bukhaari said: His hadeeths are odd. an-Nasaa’i said: His hadeeths are odd. It says in al-Kuna: He is not trustworthy. Ibn Hibbaan said: He narrates odd hadeeths from well known people and his reports cannot be quoted as evidence or written down, except for the purpose of giving an example (of a weak hadeeth). Ibn ‘Adiyy said: al-Miqdami and others narrated from him obscure hadeeths, and some of his hadeeths contain odd material.
Tahdheeb at-Tahdheeb (3/305-306).
The hadeeth was classed as da‘eef by an-Nawawi in al-Adhkaar (p. 189) and Ibn Rajab in Lataa’if al-Ma‘aarif (p. 121). It was also classed as da‘eef by al-Albaani in Da‘eef al-Jaami‘ (4395). Al-Haythami said:
It was narrated by al-Bazzaar, and its isnaad includes Zaa’idah ibn Abi ar-Ruqaad. Al-Bukhaari said: His hadeeth is odd, and a number of scholars regarded him as ignorant.
Majma‘ az-Zawaa’id (2/165)
Moreover, the hadeeth – in addition to being da‘eef – contains nothing to indicate that this should be said on the first night of the month of Rajab; rather it is a du‘aa’ that asks for blessing during that month in general terms, which is something that is valid to do in Rajab and also before Rajab.
With regard to the Muslim asking his Lord to let him reach Ramadan, there is nothing wrong with that.
Al-Haafiz Ibn Rajab (may Allah have mercy on him) said:
Ma‘alla ibn al-Fadl said: They used to pray to Allah for six months, may He be exalted, asking Him to cause them to reach Ramadan, and they used to call upon Him for six months asking Him to accept it from them. Yahya ibn Abi Katheer said: One of their du‘aa’s was: “O Allah, keep me safe until Ramadan, keep Ramadan safe for me, and accept it from me.”
End quote from Lataa’if al-Ma‘aarif (p. 148)