Whilst there are hostels in Ipanema along Barao de Torre, the backpacker area is Catete. Here you can find beds from 20-100 reais ($10-45) a night and the area is pretty safe. You’re a bit far from the nice beaches but very close to the best of the nightlife at Lapa.
Apartments in Leblon and Ipanema go for around 600-1500 reais ($250-600) and whilst the cheaper ones may not be the greatest places to stay in the world you are close to the beach.
In Copacabana the prices drop to around 300-500 reais ($140-220) but you pay for that with the noise, congestion and crime.
Elsewhere in Rio de Janeiro 300 reais ($140) is a good average of what a studio apartment might cost you. If you choose to live downtown then you should know your way around so that you don’t get mugged each night returning home.
A cool place to live for the alternative traveler is Santa Teresa, a hangout of freaks and artists that represent the counter culture of Rio.
If you’re staying for just a few weeks in Rio then you can always look into apartments de temporada, short term lets. The prices vary enormously but if there’s a couple of you then it’s definitely better than staying in a hotel.
When you do get an apartment remember to be very nice to the guys working on the front desk as security. If you have any problem they’re the first guys who might be able to help you ie they’ll tell your neighbour to cut out the karaoke at 6am each morning. These porters are often from the North East of Brazil and earn shit wages so leave them a good tip.
Also read about a traveler who lived in an apartment in Rio.