So we have now Centro and South. Both places have attractions. Copacabana, Arpoador, Christ Redeemer, Sugar Loaf, Tijuca Forest (good to go hiking in Rio), Favelas and historical sites are around here. And honestly, they are not that far from each other. South has 2 additional advantages: closer to the beach and a more active nightlife, with restaurants and bars open every day. ‘Centro’ has almost nothing going on at night, except Lapa, the very specific tiny part of Centro where there’s some good nightlife of Rio de Janeiro, but mostly on weekends.
So let´s focus on the South, which is not a very big area, but it has a lot of different neighborhoods. Some places are close to the beach and some are closer to ‘Centro’. Not within a walking distance to the coast, we can list Gloria, Santa Teresa, Catete, Laranjeiras, and Botafogo which are an older part of Rio. Cheaper in the accommodation, close to the main attractions but away from the ocean. Santa Teresa gains extra points because it´s a picturesque bohemian place and it has amazing guesthouses and boutique hotels, so consider that if it fits your profile.