In this short (as in some of the others) Superman leaps from place to place (like later Marvel's Incredible Hulk), rather than flying. This is from the original comics version, in which he didn't fly either. He was given the ability to fly, in the comics as well as the animated series when the creators realized that what looked good in heroic static drawings made him look like a red-and-blue kangaroo bouncing over buildings. See more
Up in the sky, look! It's a bird! It's a plane! It's Superman!
Faster than a speeding bullet, more powerful than a locomotive, able to leap tall buildings in a single bound, this amazing stranger from the planet Krypton, The Man of Steel: Superman! Possessing remarkable physical strength, Superman fights a never-ending battle for truth and justice, disguised as a mild-mannered newspaper reporter, Clark Kent.