Run. Keep running. Run again. Run again. Run again.
With your prior running experience, you'll likely be surprised at how quickly you're able to return to running, even with your current state. It'll be tough the first couple weeks, but (especially with the past experience) you'll be able to reach a point where a slow jog feels easy. This slow jog has a comparable exertion level to hiking; the easier you can make jogging, the better your vacation in the mountains will go. Gaining proficiency with other cardio exercises would also help, but running will provide the most benefits, both because it targets the same muscles as hiking and because you've done it before. A few weeks of running can dramatically change how light you feel on your feet; this change will help both physically and mentally.
As for how to start running again, take it slow. It's important not to try to reach too far and disappoint yourself. The feeling of wanting more after a short run is dramatically more beneficial than detesting the activity after a failed longer run. The exact distance to start with really depends on how out of shape you are, but trying for 20% of the distance you used to do, and at a slower pace, should work well. The next day, if 20% felt good, try 30%, or do 20% at a faster pace. Work your way up this way, keeping the running doable. As I mentioned before, you'll likely be surprised at how quickly you can get back into shape.