You might give Colorado Mountain Express a call 800-525-6363, www.ridecme.com. They do regular pick up at DIA. Although Winterpark is not on their usual list, I know they ll sometime deviate. You might also consider doing an overnight in Denver near the Amtrak Station (Comfort Inn, Brown Palace and DoubleTree Hotel are nearby) and taking the Ski Train (Amtrak) up in the morning. The ride is beautiful. Enjoy. I hear the snow is wonderful now.
Sorry, there is no cheap way, but there are shuttles. Greyhound only has one bus a day from Denver to Winter Park , at 11:00 in the morning. you could call the hotel where you will be staying and ask if they can recommend a shuttle service.
rent-a-car.... or taxior go to union station and take the ski train up to the resort and take a bus to winter park