The requirements of both of these two goals are the same. So you can train the same way for them: Lots of vertical days on skis. I would include 2x/week some core and basic strength training sessions as outlined in our book. But, essentially you just need to spend a lot of time on your skis going up and down. The more you do this the easier both of these goals will be. You might back off training for a week before each goal as a taper period and you might want an easy week or few days after the hut trip before starting back in building weekly vert towards the Rainier ski.
Since skinning uphill is already very ME intensive I would not recommend doing supplemental ME workouts unless you have long breaks between ski days with much vert in them. Its all base training and the more of that you have the better.