There is a lounge for Diners Club card holders that will store luggage if you have a Diners Club card. There is also an American Express Centurion lounge that is fairly nice. Both are a little hard to find. They are on the level above the checkin places before international check in. Most of the airline club lounges are after you go through international check in.

I don't believe the Diners Club or Amex lounges have showers but there may be some in one of the airline club lounges. (I am not certain about the showers in the DC and Amex lounges because I have never specifically looked for them. I don't recall seeing any though.)

You can take an airport bus into the middle of Sao Paulo and walk around, but the traffic is terrible. Count on two hours each way to be on the safe side. I had the bus drop me off at the Intercontinental Hotel which is in a great neighborhood with shopping, an art museum, parks, etc. The bus leaves the airport frequently. I am not sure how you would catch the bus for the return trip but I am sure they could tell you if you ask at the Airport bus kiosk in GRU. You'd probably have to arrange a pickup at one of the hotels. I don't remember the exact price but it was about $25 (American) each way.

BTW, there is supposed to be free internet service in the airport but it was not functioning when I was there on Tuesday. They did have internet service at the Oneworld lounge but I don't know if the other clubs have it.