For anyone interested, to expound upon Will's answer, it was discovered in 2011 that birds use something called optical flow to determine distance.
In computer vision, optical flow tracks the apparent movement of points as an observer moves through a scene. This picture tracks the position of similar points between two frames in a video. Notice that points farther away appear to not move as much. Points which don't change as much between frames are assumed to be farther away.
In the experiment, a bird was made to fly between two walls. The walls had either vertical or horizontal stripes. When moving through this hallway, the wall with the vertical stripes would move past the observer, allowing for optical flow tracking. Walls with the horizontal stripes, however, would not not appear to change as the bird flew; in the wild, this would mean that the object was far away (as we saw earlier in the description of optical flow).
It was found that the birds always flew in the middle when both walls (A) had vertical stripes (both eyes could use optical flow tracking). When one wall had horizontal stripes (B and C), however, the bird flew much closer to the wall with these horizontal stripes. Since the pattern didn't appear to change as the bird moved through the hall, the bird assumed that the wall was farther away than it was.
For those interested, the name of the paper is Optic Flow Cues Guide Flight in Birds. There is also a video which shows them flying.
tl;dr To answer the original question, objects which are closer appear to move more than objects which are farther away. Optical flow tracking would absolutely be a viable method for a cyclops (or any animal with monocular vision) to use to track distance.