Is it possible for a prey to become its main predator's predator?
Also it should be noted as you move from prey to predator, there is less energy to be had. This is why we often do not have a predator of a predator species. There simply isn't enough food at that level of a predator to support another predator above it.

That said, I am assuming you are thinking of an alien ecosystem. In which case you might be able to sustain such a system if there is a massive flux of bio-matter through the ecosystem. Ie plant food is extremely abundant and animals within it reproduce at an extremely fast pace. We can see such an environment at the bottom of a marine ecosystem in the polar regions during summer (near constant sunshine, and an up welling of minerals from the deep sea. 


  If that's not possible, is it at least possible for X to evolve
  adaptations for fighting/killing Y (Not for food)?


Happens all the time. While lions do hunt cape buffalo, cape buffalo can and do kill unwary lions.