We stayed at POP in May, and our ADR was for 7:30 in the morning, the bus driver from POP actually took us right to the Contemporary's front door, as MK still wasn't open yet. So depending on what time your ADR is for, you might get door to door service!
If you don't have an early AM ADR, do like the other posts have stated, take the bus to MK and either walk or take the monorail to the Contemporary.
Have fun!!!