The Haute Route?
Lots of sunscreen, the higher the SPF rating the better.
Stay hydrated- altitude and cold will dehydrate you faster than most people realize.
I'm told that alpine huts are great places to catch colds or other respiratory illnesses. People book a trip, come down with something just before they leave, and don't want to cancel so they tough it out. Add in crowding and lots of tired people at the end of the day and the microorganisms are just having a ball. Bring whatever you use to keep your immune system up, like loads of Vitamin C.
PaulF's suggestion is actually a very good one. I don't think you can tour for 6 days in the Alps without encountering at least one significant downhill stretch, and you'll enjoy it far more (and be a lot safer) if you can negotiate the downhill stretches with some skill. If you're already a reasonably good downhill/Alpine skier, put in some practice time on whatever ungroomed terrain you can find, be it bumps, powder or crud. If you're not an experienced downhill/Alpine skier, learning how will make your tour much more enjoyable (and it's a lot of fun in its own right).
In that vein, any aerobic sport that develops leg and foot coordination and balance will make the ski tour much more enjoyable. Soccer, trail running, ice skating, roller blading, surfing and mountain biking are all good.