There is nothing wrong with congratulating one another at the beginning of the month. The Prophet (peace and blessings of Allaah be upon him) used to tell his Companions the good news of the onset of Ramadaan, and urge them to make the most of it. Abu Hurayrah (may Allaah be pleased with him) said: “The Messenger of Allaah (peace and blessings of Allaah be upon him) said, ‘There has come to you Ramadaan, a blessed month. Allaah has made it obligatory on you to fast (this month). During it the gates of Paradise are opened and the gates of Hell are locked, and the devils are chained up. In it there is a night that is better than a thousand months, and whoever is deprived of its goodness is deprived indeed.’” (Reported by al-Nisaa'i, 4/129; Saheeh al-Targheeb, 1/490)
And Allaah knows best.