Let me make a small correction here. While some of the buses run until 2:00 a.m., the monorail does not. The monorail trains run until MK and Epcot are clear of guests. There is no fixed shutdown time. It takes between an hour and two hours for each park to clear. So if Epcot closes at 9:00, for example, you can expect the monorail to run until at least 10:00. Usually closer to 11:00.
In the mornings, the resort monorail starts running at about 7:15. The express monorail between the TTC and MK starts as MK opens. The Epcot monorail starts at about 8:15.