In my experience the transfer from mountain running to cycling is way better than the other way around. So if you are doing a lot of vert in your running training you should do OK on the bike (other than getting a sore butt). If your bike trip is on alpine passes with big climbs then toss in 1x/wk of uphill cycling ME workout. Otherwise I think you might want to add do one long ride a week if you are going to be riding for over 2-3 hours/day on this trip. All this extra training is going to mean reducing the load someplace else maybe. This depends on current levels of fitness for alpine climbing/approaching and biking though.
It is always going to be a compromise of some kind when you start mixing objectives. That doesn’t mean you can’t mix them but just understand that you will probably not maximize both areas.