Hi Stephanie,
The quickest way to get from Disney's Pop Century Resort to
Mickey's Backyard BBQ which is held at Disney's Fort Wilderness Resort & Campground using Disney's complimentary transportation would be as follows:-
1/ Take a bus from Pop Century to the Magic Kingdom and then from there take a boat to Fort Wilderness Resort & Campground.
OR
2/ Take a bus from Pop Century to Epcot and then transfer to another bus to Fort Wilderness Resort & Campground.
Both journeys will take approximately 40-45 minutes and you will also need to allow up to 20 minutes for time which may be spent waiting for each bus or boat to arrive.
If you arrive at Fort Wilderness Resort & Campground by bus, you will then need to take the internal bus (Orange Route) from the Outpost Depot to the Settlement Depot.
The Dinner Show is held in the Outdoor Pavilion which is located just behind Pioneer Hall.
As Guests are advised to arrive at least 30 minutes prior to show time, I would recommend leaving your Resort hotel at least 2 hours prior to the start of the show.
Wishing you a magical trip!