Cross back over the Ring and take a right (walking along it counter-clockwise) until you reach the lush Burggarten, on your left. 
Here you’ll have the chance to take a selfie with Mozart, meander through a verdant landscape designed by an Austrian Emperor, learn why you’re now allowed (and encouraged!) to sit on the grass,  visit the butterflies or sip coffee in a former imperial greenhouse, and catch a showing of the iconic film The Third Man at one of the world’s oldest continually-operating cinemas. 
A map of the entire route can be found at the following link: https://tinyurl.com/zxjv65n6