If it's a nice sunny day and you have an hour to spare, you can walk all the way from Stuttgart city centre through 'Castle Gardens' to within 100 yards of the stadium (alternatively see Stadium by car or rail).

The view from the TV tower above the city (not taken on a sunny day!!) gives you an idea of the distance involved.  The red square is the area in the photo below.

The trees you can see in the photo above mark the western end of the park, which starts at the Neues Schloss (New Castle or Palace).

The map above shows you how Schloss Garten (Castle Gardens) runs in a long curve out to the large area of grassland, the Cannstatter Wasen (Cannstatt Meadow), on the bank of the River Neckar.  The stadium is just across the river. For directions to the stadium by road or rail, see Stadium by car or rail.