This is really last minute, but... We are taking the train in to Zagreb tomorrow and then we have to get to the airport. We were just going to take a taxi, but didn't know if there's a simple way via public transport to get there so we can save some money...
You can catch the tram from the train station to the Central Bus Station and from there a bus to the airport.
http://www.zagreb-airport.hr/en/gm_kako_do_nas