Originally Posted by
offerendum
Careful! Normally a saunaclub is not the place to be

Unorthodox? Yes. But it does meet OP's requirement for "a shower to freshen up and change" and is not subject to availability of rooms at hotels which can be hard to find last minute during messe.
Cheapest and quickest option would be the airside lounges either LH or Hugo Junkers.
Sheraton and Maritim would be the next quickest option but I'm pretty sure they will charge more than €50 even for a dayroom (Sheraton offers dayroom not sure about Maritim).