The month of Rajab is one of the sacred months. Observing fasting in that month is a praiseworthy act that entails a great reward. The Prophet (peace and blessings be upon him) is reported to have encouraged his companions to observe fasting in the sacred months including the month of Rajab.
Stating the reward of observing voluntary fasting during the month of Rajab, Sheikh `Atiyyah Saqr, former head of Al-Azhar Fatwa Committee, says:
I would like to stress the fact that observing voluntary fasting during the sacred months or Al-Ashhur Al-Hurum, including the month of Rajab, is recommended or mandub. The Prophet (peace and blessings be upon him) is reported to have instructed his Companions (may Allah be pleased with them) to observe voluntary fasting in Al-Ashhur Al-Hurum.
He (peace and blessings be upon him) used to urge his Companions to fast three days of every month. Thus, fasting during the month of Rajab is recommended but there is no authentic hadith that indicates that fasting certain days of Rajab, for example the first day, entails a special reward.