The Museu da República is housed within the Palácio do Catete and was Brazil's presidential palace…read morefrom 1894 to 1960. It was turned into a museum after the capital of Brazil was moved from Rio to Brasília.
The outside is very unassuming and not well-marked, so you could easily walk right past it. But the inside is nothing short of amazing. Most of the rooms have been preserved with original furniture and the paintings, statues, and frescos have been kept in excellent condition as well. Honestly, I'm having a hard time describing what the rooms look like, and I'm not sure words can do them justice. This is an important part of Brazil's history and I can't recommend a visit highly enough (come here instead of seeing the painted stairs).
You can tour the old ballrooms, the dining area, the meeting area, see the former Presidential Office, and many other rooms. There is a third floor where President Getúlio Vargas killed himself (the Presidential Bedroom), but it's closed for renovations/preservation.
Well worth the visit - take exit B from the Catete Metro Station and you're right there. The entrance fee was R$6, and there are lockers where you can leave your bag or coat.