A taxi from the airport to the city center should cost 30-50 PLN (7-11 Euro).
There is just one bus 406 at the Airport
(Lotnisko), which goes to the Main Train Station (Dworzec Główny - DWORCOWA stop). You need to go a bit straight and then right to get from the DWORCOWA bus stop to the Main Train Station tram stop.
From the Main Train Station to the Mathematical Institute and Kredka
Take tram 2 direction Biskupin and
get off at the PL. GRUNWALDZKI stop (a big square, 6 stops from the train station). You can see Kredka (crayon) and Ołówek (pencil).
If you go to the direction opposite to Kredka, you will reach the Grunwaldzki Bridge and the Mathematical Institute.
From the Main Train Station to some hotels
Take tram 9 direction Sępolno or tram
17 direction Sępolno and get off at
Katedra (4 stops). Go to the cathedral, pass it and after 100 meters you will be nearby
John Paul 2 Hotel.