Hi Maria:
Buses start running early to the Magic Kingdom for Guests who have early morning reservations at restaurants inside the Magic Kingdom and close by Resorts. These buses are unofficially referred to as "breakfast buses." These early-morning breakfast buses begin to run from Resorts as early as 6:30 a.m. so you can make your early dining reservations.
I would plan on waiting for your bus by 6:30 a.m. The bus will take you to the Magic Kingdom, and you can take the walkway (about 10 minutes) over to the Contemporary where Chef Mickey's is located. Inform the bus driver that you have reservations at Chef Mickey's. During early morning bus routes, it's possible the bus driver may be able to drop you directly at the Contemporary Resort for your reservation. Of course, if you are anything like me and want a few extra minutes of sleep, peace of mind, and to guarantee that you will be at the restaurant on time, you can take a taxi. Taxis are generally waiting outside Resorts for passengers, but your Resort concierge can help you arrange one.
Complimentary Resort transportation runs
an hour after Park closing. I would show up at Tusker House a few minutes before your reservation and note to the host and your server that you are relying on Disney bus transportation to return to your Resort after Park closing. The host and your server should be able to work with you, if you work with them (e.g. you may want to save lingering over a cup of coffee for a different night), to get you to served quickly and to the bus in time.