The only facts known about Jeannie's birthday were that it was about 2000 years ago and that Neptune was in Scorpio on that day. The existence of the planet, Neptune, was not known before its discovery in 1846, about 120 years before the events depicted in this story. Therefore, nobody would have known about either Neptune or its position in the sky at the time of Jeannie's birth.
At any point when Jeannie was in the desk drawer, Tony could've simply put her in his pocket and taken her home.