The drawback of Aeroflot flights to and from Kazan is their late arrival and early departure. To take a taxi to/from the city costs something, and also you will not be able to sleep well. The good option is to overnight in the vicinity of the airport. The hotel Polet (Flight) is just 500 meters away from the main entrance. The prices are very reasonable (from about 10 usd per person up to 100), but the rooms are very different. The cheapest option may be terrifying, but if you used to live in former USSR you would not be embarrassed by leaking tubs and falling ceiling. The most important is that bedclothes are clean and windows can be closed tightly. More expensive rooms are much better, but the number of rooms is limited, since this hotel is used by flight attendants and pilots.
We stayed there twice (after arrival to Kazan and before departure).
To get to the hotel from the city center one can take a regular bus 97 (ride - 1 usd)
Plus: price, location, staff is good.
Minus: old not renovated rooms