you won't be running on rainier, unless a tauntaun is chasing you, so...
as long as you're in good enough shape to ascend 9000 feet in 2 days w/o stopping more than once an hour or two and not being a total piece of jelly at the end of it you'll be fine, and that's pretty easy to test below timberline
running is useful for developing mental hardness, but i suppose you could just jerk off w/ sandpaper for a few weeks and get the same results
i assume this'll be your first time on rainer - it's a fantastically cool place - enjoy