We had a lovely stay here. It is indeed right behind Schonbrunn so we enjoyed walking down every day through the park. The downside that the park closed at 17:30 so after you'd have to walk up the hill without the view. However, there is a bus that leaves you right near the house which is great.
We had a bit of a problem with the internet but the hosts were great and promptly fixed it.
The house is beautiful, clean and has all the amenities you need for your stay.