Tough assignment - Astana has to be one of the most cold and depressing places I frequent (and I do get to many cold and depressing places)
No sign of a GA scene there that I could see, though I bumped into a couple of British ATCO's in some random bar in Almaty who were out there on a training mission. They said flying there in a "G" reg on my UK licence was possible provided I had a Russian speaker on board.
Not much help sorry but that's all I know (PS. Hope you like horse testicles, a Kazakh delicacy
)