Moscow's "sport flying" airfield used to be Tushino - a PPL I was on a trip with in the late 90's managed to get a dual flight in a Yak-18T there, after about 2 hours of negotiation through an interpreter. Sadly, Tushino is now closed.
On a visit last year, there were a few light aircraft (including Cessnas and Robinsons) at Bykovo, so that might be the place to enquire. Good luck!
Rhys.