Is there a bus which connects the airport to the city centre?
Yes, it's called the Conexão Aeroporto. There are two options, both to downtown. One goes to the downtown bus station (Terminal Rodoviária) for R$12,70 and the other to the Av. Álvares Cabral (officially in what's called the Lourdes neighborhood) for R$27,70. The more expensive option is in a nicer bus, with airco. Both options are safe.
yes. The company that handle this transportation is called "Conexão Aeroporto". And you pay about 30 reais to use it