This possible error fare deal with Alitalia is available in travel period within April and June 2015.Max stay in Brazil 3 months. All these European cities are easy to reach either by bus or low cost airlines. To check your connection possibilities from your home town directly you may check through Skyscanner. Cheap accommodation in Rio de Janeiro or Sao Paulo is possible to compare and book here.
Booking samples
Budapest - Brazil from €354
Germany - Brazil from €377
UK - Brazil from £280
Brussels - Brazil from €391
Amsterdam- Brazil from €390
Booking sample of cheap return flights from Europe to Brazil for €354 here. In this case you would be flying from Budapest to Rio de Janeiro on 7th of June and returning from Brazil back to Europe on 20th of June 2015.