I am trying to find a way to check the public transportation offer from Fraknfurt Hahn airport to Koenigstein (Taunus) but it seems that this data does not exist in public search engines.
Any ideas on where to search?
From the zip code you gave (=Postleitzahl) you want to reach Koenigstein im Taunus. This is accessible from Hahn airport, even though it is not the easiest and fastest trip.
I could search it on the Deutsche Bahn and found a possible connection. You have to take the Bus 600 to Frankfurt Main station (Frankfurt Hauptbahnhof in German, sometimes shortened in Hbf). The ride is quite long, about 1h45. Then you can take a train from Frankfurt Main Station, Koenigstein im Taunus being the last stop of that train (apparently the train often leaves from platform 22 or 23). The ride is less than 30 minutes but the connection can be longer, there is apparently one train per hour.
Regarding the price I could not find information though.
Note: also note that while the bus 600 from Hahn stops at main Frankfurt airport which is better served by public transportation, stopping there is not a better alternative as it takes 1h30 for Hahn-Main airport.