How should Viennese coffee be taken?
Based on your picture, you're asking about Weiner Melange coffee in Vienna, a serving of half espresso coffee and half milk foam.  That's instead of American "Viennese Coffee", made with whip cream and cinnamon.  Assuming that's correct:


The spoon is intended for mixing the milk foam and coffee.  You may wish to add sugar as well.
The sparkling water is a chaser to clear your palate.


The above is based on personal tourist experience and some internet research.  Some links:


https://en.wikipedia.org/wiki/Wiener_Melange
http://www.wisegeek.com/what-is-a-wiener-melange.htm
http://www.tasteofaustria.org/coffee/