How far back man began to domesticate animals remains still to be solved. I would be inclined to think that such domestication probably began long before we know.
Wherever man or Neanderthal lived they left scraps from their meals that would have attracted mice and other animals. The work of a cat on these small vermin would have been noticed by the humanoid, not to mention the advantage the cat would have hanging around the humanoid.
Mutual domestication I suppose.