I have a flight from Kazan to Moscow that departs at 09:00. How much earlier do I have to arrive at the airport for an inner-Russian flight (I hold a tourist visa) and what is the best strategy to get to the airport from Kazan city center?
If the airport has a website it could be your best friend.
In this particular case Kazan International provides this information (in Russian and English).
Under section "For Passengers and Guests" there is a section of How to Get there with section specifically by Bus. It lists that there is a bus №97 which starts running at 5 am, though reading some of the reviews on the bus site (in Russian) it could be running every 30 minutes, it's inconsistent, and the first one may not even arrive.
So if you want reliability I would suggest grabbing a cab though it is much more expensive 850 vs. 39 Rubles.