The musty smell is from mold and mildew which can be killed with bleach. However, they also can cause many illnesses and respitory propblems for people who are close to it and breathe the spores.
I don't think you can get it out of a mattress but I suppose you can try any of the suggestions given. Call your health department on this. Best bet may be to toss this old mattress and buy a new one. An area health department recently condemned an entire house cause the mold and mildew was in the walls and floors. The family had major health problems for years and didn't realize what the cause was.