Is “Allahumma Ballighna Ramadan” Authentically Reported?
Question

 
I would like to know whether it is Sunnah to say a particular du‘aa’ on the first night of the month of Rajab?
The du‘aa’ is as follows: “O Allah, bless us in Rajab and Sha‘baan, and let us reach Ramadan”.

I ask Allah, may He be glorified, to make us steadfast in acting upon the saheeh (sound) Sunnah.
Praise be to Allah.Hadiths on the virtues of Rajab
There is no sahih hadith from the Prophet that speaks of the virtues of the month of Rajab. 
Ibn ‘Uthaymin (may Allah have mercy on him) said: 
“There is no sahih hadith that speaks of the virtues of Rajab . The month of Rajab is no different from the month of Jumada al-Akhirah 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.” (Liqa al-Bab al-Maftuh, 174/26) 
Degree of the hadith “Allahumma Ballighna Ramadan”
It was narrated by ‘Abdullah, the son of Imam Ahmad, in Zawaid al-Musnad (2346), at-Tabarani in al-Awsat (3939), al-Bayhaqi in ash-Shu‘ab (3534) and Abu Nu‘aym in al-Hilyah (6/269) via Zaidah ibn Abi ar-Ruqad who said: Ziyad al-Numayri told us, from Anas ibn Malik 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‘ban, and let us reach Ramadan.” (Allhumma barik lana fi Rajab wa Sha’ban wa ballighna Ramadan).
This isnad is da‘if. Ziyad al-Numayri is da‘if and was classed as such by Ibn Ma'in. Abu Hatim said: He cannot be quoted as evidence. Ibn Hibban mentioned him in ad-Du‘afa and said: It is not permissible to quote him as evidence. (Mizan al-I'tidal, (2/91) 
Zaidah ibn Abi ar-Ruqad is more da‘if than him. Abu Hatim said: He narrated from Ziyad al-Numayri, from Anas, odd marfu‘ hadith, which we know only from him or from Ziyad. Al-Bukhari said: His hadiths are odd. An-Nasai said: His hadiths are odd. It says in al-Kuna: He is not trustworthy. Ibn Hibban said: He narrates odd hadiths 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 hadith). Ibn ‘Adiyy said: al-Miqdami and others narrated from him obscure hadiths, and some of his hadiths contain odd material. (Tahdhib al-Tahdhib, 3/305-306) 
The hadith was classed as da‘if by an-Nawawi in al-Adhkar (p. 189) and Ibn Rajab in Lataif al-Ma‘arif (p. 121). It was also classed as da‘if by al-Albani in Da‘if al-Jami‘ (4395). Al-Haythami said: 
“It was narrated by al-Bazzar, and its isnad includes Zaidah ibn Abi ar-Ruqad. Al-Bukhari said: His hadith is odd, and a number of scholars regarded him as ignorant.” (Majma‘ az-Zawaid, 2/165) 
Moreover, the hadith – in addition to being da‘if – contains nothing to indicate that this should be said on the first night of the month of Rajab; rather it is a du‘a that asks for blessing during that month in general terms, which is something that is valid to do in Rajab and also before Rajab . 
Can Muslims make the du’a “Allahumma Ballighna Ramadan”?
With regard to the Muslim asking his Lord to let him reach Ramadan, there is nothing wrong with that. 
Al-Hafiz 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 Kathir said: One of their du‘as was: “O Allah, keep me safe until Ramadan, keep Ramadan safe for me, and accept it from me.”” (Lataif al-Ma‘arif, p. 148) 
And Allah knows best.