I have got my H1B approved and my OPT expires 1 month before my H1B starts. What is my status going to be during that one month period? Can I still work or do I have take a gap of 1 month before my H1B starts?

You cannot work unless you have OPT. if your H-1B is approved, go to the your DSO/FSA and they will enter into SEVIS and they will get a new I-20, which will be annotated to show the employment authorization to cover the cap gap good luck.