Posting ahead is pretty common for cyclists in the US. I've never heard of any problems with General Delivery.
However, since you are shipping a bike box, I would suggest you use UPS or FedEx, rather than the Post office. I would also suggest that you have your box sent to a hotel. Durango is a fairly small town, so your choices may be limited, but contact the hotel ahead of time, and make sure they will hold your box until your arrival. The hotel will tell you how to address the shipment.
Both FedEx and UPS will pick up your box at your hotel of departure in Jasper, for a small fee.
For cost, I'm going to guess $50USD. Expensive because of size not weight. Log on to either site, and you can get a cost estimate.
Enjoy your trip!