Hello,
I used to live in Geneva.... There are public, coin operated showers in the baggage claim area, after they check your passport but before customs. I have no idea how much they cost, but believe it is 5-10chf. Though I have never been in one, I would assume they are as spotless as everything else in Switzerland...
However, I do not think there is any way to access the airport lounges on the arrival side. Upon arrival, you may be able to go up the stairs before passport control, which dump you out in the airside shopping area, but I know for a fact the Swiss Business Lounge, BA Lounge, LH Senator Lounge do not have shower facilities. Swiss First might, and there is a Priority Pass type lounge as well, but I have no idea about them...
Good Luck!