There are not any 'character' buses anymore. BUT...the buses to the parks do start running arounn 6:15ish for those very early morning breakfast ADRs. So, here's what you need to do.
For early ADRs inside MK, or at the CR, Polynesian or GF....be out at your resort MK bus stop at least an hour before your ADR time. When you get on the bus, ask the driver what the quickest way to your particular resort is from the MK. You may get lucky and the driver can take you right to that resort. If not, then you can get off the bus at MK and walk to the CR (about 8 mins), or take the monorail to Polynesian or GR. The monorail starts running about 7ish.
It's a bit tougher getting to the other breakfasts. But, it's a good idea to head to a nearby park an hour or so before your ADR and then transfer to the resort bus you need.
Or you can just take a cab. I generally do not make really early breakfast ADRs unless I can walk to the resort (Cape May Cafe when I'm staying at BW, S/D or YC/BC for instance) or I can get right there on a park bus and then head to the resort (CR, Poly, or GF).
