Picking up from getting to the bus/train station in my last post. Don't worry, there's only one station in Jönköping.
You can ask inside the station either at the Tourist Info or any employee how to get to the University, but it's actually really close and easy. I'll draw a small map below and describe here in words how to get there. Assuming you arrive at the bus station during opening hours walk around and find the Pressbyrån. Next to it you'll find and take the escalators up. You'll go through a "tunnel" with different offices like the bus office (Länstrafiken) and just walk out to the street. You'll see to your right many bus stops (this is the main city center bus stop - Juneporten). You'll want to walk to the left (right as we see it on the map below) until you reach the street Kyrkogatan (gatan means street in Swedish) and you'll want to turn right onto this street. You'll see a park on that corner before turning right. Then you'll walk straight as you can see on the map and directly reach the university. It's pretty much that easy.