There are two good ways getting to and from the bus station in Rio: One is The REAL Premium bus and the other is taxi. There are also options with local buses and also subway and bus integrated, I don’t think they are worthwhile, either it’s not safe or it´s to much of a hassle.
The REAL Premium Bus
The price is 10 R$ and it runs everyday from 5:30h until 21:30h at least every 20 minutes. Note that where there are designated bus stops like in Copacabana and Ipanema, you can only catch and get off at the ones named BRS 1. It’s not fast (45 min – 1h 30 min depending on traffic) but it’s reliable. Click the link REAL Premium bus for updated info from the company site.
Going from the bus station in Rio you can easily catch it by going outside the arrival terminal to the left; the REAL bus leaves from there. There are two different routes, one is to Leblon and run everyday, the other one is to Gávea and run weekdays, both of them pass Zona Sul. On Sundays and holidays between 7h and 18h (during Daylight Saving Summer Time until 19h) they shut part of the beach avenue in Flamengo and the bus going to Leblon changes route from Aterro to Praia do Flamengo.
Going to the bus station in Rio the REAL Premium bus more or less reverses the routes, note that these routes are not affected by Sundays or holidays.
Going from the bus station in Rio there is a not so reliable taxi cue system on the left outside the arrival terminal, you can go for fixed price or on the meter (it ends up about the same). Avoid the taxis you are offered inside the arrival hall, they are more expensive.
Going to the bus station you can order a normal yellow cab from a company by phone (need to speak Portuguese), they will go on the meter; If they don´t, tell them to turn it on. Another option is to flag one down on the street, it’s easy to make them go on the meter or you can always try to negotiate, but it’s a bit harder in my view. Click Taxis in Rio to get more detailed information on this subject. Uber is an option that has grown a lot in popularity.