Fortunately from the Axamer Lizum webcam there's already snow base, and they're making snow too. In addition, it's at altitude, so there's an added advantage for colder temperatures.
However the start of the season is always weather dependent, and there it's usually mid/late November. So it all depends on which runs you want to do, and luck, and how late you can leave your trip. Best you can do is keep an eye on the snow report and weather on their website and keep any eye on the webcams as well. Their lift page should also start to indicate which runs are open as they get enough of a base.
You can also check out the climate for Innsbruck and note the snowfall for that time of year. Check out weather sites too and see if there's snow falling on the long rage forecasts. It should give you a decent idea.
Finally you can check on instagram or flickr or other photo sharing sites when it opens. If people are taking GPS-tagged photos, you can get an idea of the view from their shots and how beautiful different runs might be.