Railway track is intended to last around 100 years according to this study, however that is for heavy locomotives and rolling stock travelling at their full design speeds. Some rails may degrade and be replaced in as little as 30 years with corrosion and heavy use.
However, we're talking about a handcar here, not a heavy locomotive. Since we're talking about a light, slow vehicle that applies very little force to the rails, rails such as "e" in the following image could reasonably be expected to be useable by a handcar This may be as much as a couple of hundred years old. It would be a rough ride, and it might be necessary to lift it over particularly bad sections of track, but depending upon metallurgy, the rails may be two or even three hundred years old and still be useable... for that value of useable.
Don't even think about putting a full-sized train on it though. It will derail.

EDIT:
Since the OP wants a rate of corrosion per year, this abstract shows that carbon steels (from which railway rails are usually made) in water may corrode at around 10-100 μm/year, averaging 50 μm/year. In a 'rocky mountains-like' area, we wouldn't expect great amounts of salt which tends to accelerate corrosion.
If we consider the profile of a typical railway rail:
(Image from https://en.wikipedia.org/wiki/Rail_profile)
Corrosion tends to happen where water beads on the metal surface. Water on these rails would be falling on the head and foot of the rail, and would tend to run more quickly down the web of the rail. As can be seen from the photos of corroded rails, it is the underside of the foot of the rail where the most corrosion occurs, mostly where the rail contacts the sleepers. The head of the rail would be most exposed to sun and wind, which would tend to dry the rail more quickly after rain.
As I have previously said, while a rail may be corroded to the point that running a heavy train on it would lead to derailments, this would probably occur on a bend where the centrifugal forces push outwards against the rails and cause failure of the corroded rail foot. On straight sections of track, rather large failures of the entire rail are required to cause derailments, as can be seen in https://en.wikipedia.org/wiki/Rail_sabotage, particularly the video https://en.wikipedia.org/wiki/File:Derailment_226-b-6082.webm, which shows the considerable lengths that saboteurs must go to to successfully derail a train on a straight section of track.
Since the OP has stated that a handcar will be used, we can expect that even an extremely corroded rail will be able to support such a light vehicle for hundreds of years past the point at which a heavy train would inevitably derail.
In Are stainless steel railway tracks worth it?, I asked if stainless steel rails would be practical, as these might last unused in the weather for thousands of years or more, but the consensus appears to be that for a working railway where track wear is an issue, stainless steel is too expensive and doesn't last long enough, and regular high-carbon steels will probably be used. However, only the OP can say if these tracks are intended to be low-use, low-maintenance stainless tracks, or high-use, regular-maintenance high-carbon-steel tracks.