Of course you can walk to the slope transport system, though I don't remember how far it is. I walked from the center of town to the gondola up to Eggli, but, again, don't remember how far it was (edge of town . . .).
Here's a link to an interactive map of Gstaad & its trails. It may help you visualize the distances:
You can click & enlarge different sections of the overall map.
Also want to suggest you e-mail (or phone!) your hotel and ask. There's probably also a skibus.
Best of luck!
Hotel Olden is in a good location - but it's not exactly walking distance to the slopes. Don't get me wrong - it's not far when you're not stomping around in ski boots. I stay at the Arc-en-ciel, and it's only a 5 - 10 minute walk (depending on how slippery the road is) - my hotel is located right next to the Eggli. I was there just a couple of weeks ago, and I did see a bus regularly coming and going. No idea how much it costs, but I'm sure it's pretty cheap. Some hotels also have thier own transportation. Gstaad just isn't a ski-in, ski-out kind of place.
The Olden is close to the end of the Pedestrianised zone, not too far from where the busses seem to stop.
Btw, Bernie Eccleston (Mr F1 himself) is part owner of the Olden.
We stayed in Gstaad this past December and had a wonderful time. I believe Hotel Olden was just across from the main drag from the train station. There is a ski bus that runs from the train station parking lot to all of the ski slopes. Keep in mind, Gstaad is tiny so even if you are not right by train station, it is just a short walk.