Hard mattress at dorm. Where to get a cheap softer mattress?

Hi there.

I just moved into my dorm in Kanazawa and the mattress on my bed is almost as firm as sitting on the floor. My body is heavy and I have a bad back, so this is just a no-go for me. Is it possible to find a cheap and maybe a rental service? The bedframe is just one big plank of wood. (Like a table.) So I would need a mattress that can take my weight, and yet support my back as well. My bed at home is a medium firmness I think. Springs of 400 pieces or so. Im not sure since my bed is old (6 years old)

I hope someone can help. My budget is about 20.000 yen.