I would agree with a taxi to DTD if you don't mind the expense. I'm not sure what time the first bus leaves Pop for DTD, and it may stop at Typhoon Lagoon on the way.
Buses from DTD to the Contemporary start running about 11 a.m. Before that, your best bet is a bus to POFQ, and transfer there to a Magic Kingdom bus. The bus would show either "FRENCH QUARTER" or "DHFQ" on the marquee. Or, you could ask the transportation cm in the orange vest for assistance, but it may not be the fastest route