From experience, the monorail starts running when the park opens.. It runs for MM, but only for disney hotel guests. You will need to show your charge slip to enter the monorail turnstile.
We have had good and bad luck with riding the monorail for park opening.. There have been times when it would have been better to wait in line at the main gate than waiting in line for the monorail..
If we are one of the first few people in line, we will wait for the monorail, if not then we will hike it down to the main gate.