The State Univeristy (Naxcivan Dolvet Universiteti) is located between the Weeping Mother Monument and Ganclik Shopping Center. The university is currently building and remodeling many structures. While the university is not off the beaten path, there are cliffs (granted not a great cliff, but I would not want to fall down it) at its far edge which offer some great views of storms over the Araz River and Mount Ararat. Furthermore, the cliffs are not commonly frequented by the locals, let alone tourists.
Getting there is easy. Get to the university. Walk up to the big blue building "Kitabxana" (Library). Take a left at the Kitabxana, and go up several flights of outdoor steps. There will be a building (currently under construction as of 7/2007), a monument, a fenced soccer field and a gym to your left. On the right will be an opened court and two instructional buildings. Directly behind the gym are some dirt paths that lead to a watershed and to the Weeping Mother Monument. Get to the watershed and you will have some great views of Araz and Ararat, along with the road leading to Turkey and the village below.
If you go back and follow the cliff, you will eventually come to the University's Music Conservatory, which I found fascination in as being an example of Soviet-style architecture from the 70s.