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Local taxi cabs take you from airport to city center, between 13 and 20 manat.
Most of city bus you can pay by cash, 30 coins (qepik) ,some of them accept card-BakiKart ( you can buy card and top up it at metro stations and bus stops. Guests have options, of paying 2 AZN for a reusable card or paying 1 AZN for disposable card, valid for 4 journeys.
Also you can use bus from airport to city (train station) it will take 45 minutes, services depart every 30 minutes, between 7 am and 11 pm. And costs 1,30 AZN.
The city also has a metro system. The fare is also like a bus, 30 coins (qepik) per trip.