What is the purpose of the "Mirror-Thing" over the Mig-29 Cockpit?
Additional details to Marco's selected answer.

On the two-seater version, the instructor seat (aft) is not high enough, contrary to other training aircraft, to see the runway. A forward view mirror replaces the rear view mirror and allows the instructor to see ahead. The runway is projected on a second mirror inside the cockpit, creating a full periscope system:


Photo by Pavel Vanka. Modified.


(Source)

Note how the projected part of the image fits very well into the direct view of the exterior.

An opening is created in the canopy when the top mirror is deployed. From below this is what it looks like (not exactly the same model):


Photo by Burkhard Domk. Modified.

The bottom mirror unfolds down to instructor eyes height:


(Source)