There is a regular bus service (Line 27) between the airport and the center of Jönköping. The journey takes about 20 minutes. Fee according to Länstrafikens traffic's tariff.
For further bus to Elmia, change to line 1 or 18 at Juneporten in Jönköping City. Please check the timetable at Jönköpings Läntrafik's web page, jlt.se, when times are subject to change by the bus company. The buses will not wait for late arrivals.