I don't know how long it took to clear through customs, but if you assume she had 6 days to make the trip, with the railroad, I think it might have been possible. From NY she would probably have gone to Chicago, then West from there, probably on the Northern Pacific Railway. Assuming the train travelled about 40 mph, that would be about 960 possible in a 24 hour interval. Adjusting for stops and layovers, adjust that down to about 600 miles per day. There is also the Canadian Railway that would have skirted just north of Montana. I just looked at a website that said the Trans Canadian Railway was completed in 1885 and averaged 24 mph from Montreal to Vancouver (476 miles/day); but that had to go across the entire Rocky Moutains and most of the trip to White Sulphur Springs would have been across the plains. Looking at
http://en.wikipedia.org/wiki/Northern_Pacific_Railway it shows the route of the railway. The problem would come in getting from Chicago up to Minnesota, where the Northern Pacific started. She could have gone west from Chicago and gotten off at town along the Mississippi River then taken a paddle wheeler up to Minnesota. Of course, this is all just speculation. She would have been one tired lady when she got off the train in Montana.
To digress, wasn't there a John Belushi movie in which his love, an eagle naturalist, had to catch the railroad at White Sulphur Springs?--"Continental Divide".