The Cedaz Shuttle Bus no longer exists. The closest thing available is the 60 Kn bus; get tickets at the kiosk opposite Exit F in Terminal 1 which is about a five minute walk from your exit in Terminal 2. This bus gets you to the train station and once there you need to enter the station and exit on the lower level to get to the taxi stop Rick recommends. Fare to our hotel in Old Town was 250 Kn which was announced beforehand and honored at the end.
Yes the Cedaz shuttle ended a few years ago.The AE bus is quite good and you can pay on the bus but don't use high value notes or credit cards.The Public Tansport from the airport is quite good and several options available depending where in the city you want to go but the 119 bus to Nadrazi Velesslavin (New metro/bus/tram hub) is the best bet as you can then easily get a metro to the city centre or tram to several places in the city.
Taxi or private shuttle from the airport to the centr will cost in the region of 600kc.
There are ticket machines in the arrivals hall and at the bus stops to get tickets for the public transport ,they take both cash and credit/debit cards.