To my russian friends, are there gyms which I could buy day or week pass?
Sure. Where do you stay in Moscow?
All Moscow gyms offer a one-time visit.
A little trick. You can ask the administrator to give you a "test day" for acquaintance with the club. Usually it's free.
@Schonefeld I'll be very close to Kievskaya station.
@Dmitriy Z I understand, but it's a bit more than that... Like you pay for one week, or for a day pass everytime you go to the gym. I will be in each city for 5/6 days so a weekly pass would be perfect.
In the post #4 Dmitry has already given you a nice list to loo through. If you stay in a hotel and they have no on-site gym, you can ask a front desk for recommendation, they may have an arrangement with the nearest gym to use for their guests.