While the beginners course of the Dutch alpine society requires good fitness, you don't have to be a strong athlete. Basically, all sorts of aerobic exercise are good preparation. If you are already in reasonable shape to begin with, three months should be enough to get ready, so September is no problem. You could easily go in July already.
Running outside has the added advantage of toughening up your feet, which I personally find helpful in preventing blisters when I go out in the mountains. If you choose something else than running, you can still get the same effect by regularly walking a couple of hours.
If you have no experience with running, make sure that you start slow and don't overdo it. Especially if you are already reasonably fit, you may find that you can already go jogging for a good distance three times a week - but then, before you know it, you get shin splints or some other problem and can't run for weeks. There are many good resources on the web regarding running for beginners, as well as explaining the dangers of getting injured by running too much too soon. The main rule is that you stop running when you feel pain.
Fortunately, there is no objection to mixing it up with other forms of exercise, so, while getting your legs accustomed by running relatively short distances at first, you can make your heart stronger by cycling, swimming or by using the cardio machines in the gym. My favorite is the cross trainer. Also, the treadmill in the gym is easier on your legs than running outside, and much less likely to get you injured.
You're not allowed to run up the Dom tower. Apart from that, running up the stairs of any high building is a fine way to exercise, but boring if you do it regularly if you ask me. And if you run down as well, it will be murder on your knees over time. You'll need those for many more years!
Climbing experience is not necessary at all for the alpine beginners course. It certainly does no harm, but the course starts from zero.
Good luck, Rob