As long as you are not hurting anything, stairs with a pack are great. The DC/Emmons routes on Rainier are walking up a hill, so if stairs are as close as you can get, great. 20 minutes at a time is another story. An hour at a time is much better. Be sure to get some long days (6 hr+) before you go. I'd also add in some bodyweight exercises, push ups, sit ups, pullups as often as possible. What training method you use is not as important as safely getting in the best shape of your life.
You are going to be climbing for 12 hours a day for 2 or 3 or more days. In April, you can expect full on winter conditions. IMO, 20 minutes of stairs with a light pack along with some light cardio will not cut it. You need to be training friggin hard. Check out some of the guide services and see what they recommend for training, check out www.mtnathlete.com
, even read this chick's comments...http://trainingformtrainier.blogspot.com/
As long as you don't get hurt, you can't "overtrain" for rainier (As in "darn I was in too good shape and it was too easy...never happens).