I second the Feira de São Cristóvão idea. It´s a very fun place and a great location to taste some of the northeastern flavours in Rio.
However, I don´t think that the Piscinão de Ramos is a great option. It´s an artificial pool/beach in a an area that isn´t easy to get to, specially if you don´t speak any portuguese.
If you want a beach far from the most touristic ones, try Prainha and Grumari at Recreio dos Bandeirantes Neighbourhood, or even Joatinga, in the beggining of Barra da Tijuca.
I prefer ten times Lapa than Vila Isabel... There are endless options in there, from the cheaper to the most expensive ones, just walk around and find something that appeals to you.
Circo Voador and Fundição Progresso, two famous venues in Lapa, will, definitely, have a special line up during the World Cup, check their websites for more info or ask in your hostel/hotel.
I also love Santa Teresa... Try the Feijoada at Bar do Mineiro once you´re there.
For an spectacular view of Rio, go to Mirante Dona Marta, located on the top of the hill in Botafogo. It used to be a dangerous place, but nowadays it´s completely safe.
The "nearby rainforest" is located everywhere, because Rio is surrounded by it. A guided tour to the top of Pedra da Gávea might be an unforgettable experience. The view from there is breathtaking. Besides, you could try a guided tour to other trails in Floresta da Tijuca (Tijuca´s Rainforest).
If you´re attending a game in Maracanã, try the Picanha in Garota da Tijuca, a traditional restaurant of the neighbourhood, before the game. It´s amazing and it isn´t expensive.
It´s located in a square named Praça da Vanhargem, also known as Buxixo. It´s located in a walking distance from Maracanã and you will find a lot of bars and restaurants in there. It´s a fun place to go before or after the match.