Proper short-term apartment rentals are not common in Brazil, but I assume during the cup you'll see more of them. The ideal situation would be if you know a local who has a friend-of-a-friend who can rent you his or her place. Expect prices to be exorbitant, as São Paulo already has very bad housing shortages. You can look for "flats," which are just hotels. They sometimes have a kitchenette-type kitchen.

Also bear in mind airfares will almost assuredly be stratospheric. I would not plan on being able to fly to the various venues. Bus travel or even car rental is viable for many cities in the south, but buy your tickets as early as possible. Brazil does not have the infrastructure to deal with itself on a normal day. The Cup and the Olympics are going to... stress the system a bit.