How did they get this shot in Metropolis?

Fritz Lang was a perfectionist and the film was considered state-of-the-art regarding special effects. I have so far found nothing about that particular scene, but did find a few thing sthat could have worked.

From the Metropolis wikipedia article, I found this reference to a camera on a swing

Among the effects used are miniatures of the city, a camera on a swing, and most notably, the Schüfftan process,[15] in which mirrors are used to create the illusion that actors are occupying miniature sets.

From this article on camera movement techniques, I found a reference to cranes and dollies being used in this time period.

However, cameramen eventually became more adventurous. Cranes and dollies were widely used by the 1920s, and the steadicam became a standard item by the 1960s

I haven't yet found a definitive answer. When I saw the reference to a sing, I thought the swing may have been used, since the scenes are short. But they last about 4 seconds and the camera movement is straight, not curved.

My original assumption, and what I strongly suspect was used, was a camera mounted on a crane being moved along a dolly track beside the runners. This could allow the camera to be above and in front of the runners, but the tracks would be out of frame to one side.

I will continue to look for something definitive and update the answer if I find it.

They are chasing after an automobile. The camera is mounted on the back of the automobile.

You can can tell that this is the case because the frame jumps around due to the motion of the automobile on the slightly bumpy road.