Depends on your school's dual degree program (ie you have to check yourself)... if so, you won't be changing subjects, really more like adding on a class(es)... I can't think of any (good) MPH programs off the top of my head that you can start with online/self-study courses... my friend at Hopkins once mentioned they were moving towards more online classes, but I don't believe you could do the MPH work off-site. Anyone else know anything about this?
And don't worry about the bolus of stress in med school. Either 1) you'll realize it's all futile anyways and relax, or 2) you'll fry your brain freaking out so much and lapse into a coma... either way you'll get through med school just fine.