I believe it. Anyone, self included, who's spent much time in the mountains knows how easy it is to get turned around. Many times on elk hunts I'd make a mental note of a tree for a marker only to find later that it looked exactly like hundreds of others in the area. I could easily see where someone could lean their rifle against a tree while they gutted or drug a deer, then couldn't find it upon returning.
Of course nothing says it had to have been there since 1882 either. I think one way to get an idea of how long it had been there would be to extract the ammunition from the magazine.
The area is quite arid averaging less than 14" of rain per year. This and the low humidity could easily explain why the metal was in no worse shape than it was. Regarding the tree against which it leaned, I did a little research and found this info:
The Utah Juniper (Osteosperma) has a long life span relative to most other plant species and a slow growth rate. At maturity, the typical Utah Juniper (Osteosperma) will reach up to 26 feet high, with a maximum height at 20 years of 25 feet.