On the month of Rajab
It has been mentioned by Abu Nu’`aym in his Hilya from Anas Ibn Malik 
said, that the Prophet of Allah 
used to make the following du`a, when the month of Rajab came in:
said, that the Prophet of Allah used to make the following du`a, when the month of Rajab came in: Allahuma Barik lana fi Rajaba wa Sha`bana wa ballighna Ramadan - O Allah bless us in Rajab and Sha`ban and let us reach Ramadan.
It has been mentioned by Abu’l-Fath ibn Abi ’l-Fawaris in his `Amali and Imam as-Suyuti has also mentioned in his Jami` as-Saghir from Hasan al-Basri that the Prophet of Allah said:
said:Rajab is the month of Allah. Sha`ban is my month. Ramadan is the month of my Ummah (community).
From the above hadith, it shows that it is preferable to make du`a, to live through the coming months, in order to perform good actions in them, because a believer increases his goodness with age. The best of people is the one who lives long and performs good actions.
The pious predecessors used to prefer dying after performing good actions, like the fast of Ramadan or returning from Hajj. And it was said by them, ‘whoever dies in this way, is forgiven.’
There is no doubt that the month of Rajab is the key to the opening of the months of goodness and blessing.
Abu Bakr Warraq said:
Rajab is a month of cultivation, Shaban is month of irrigating the fields, and the month of Ramadan is a month of reaping and harvesting.
The summary is, that the month of Rajab is a sacred month and the first, virtuous and noble of them all. So it is appropriate to celebrate it entirely and to give attention to it by performing good acts, and refraining oneself from sins and offensive things, because it is the month of Allah.
It has been mentioned that Allah said:
Fasting is for me and I will recompense one for it.
Fasting in the month of Rajab prevents one from sins, let alone preventing one from killing and having enmity towards others (for which this month was made sacred), like it was practiced by the people of the era of ignorance (jahiliyya).
(Imam Abdul-Ghani al-Nablusi, Fada’il al-Shu’hur wa’l-Ayyam (Virtues of Months and Days) pg 27, 32)