SDU (Santos Dumont, downtown Rio) has been mentioned. (I don't think airports come much closer to the centre than that), and there's Congonhas which is definitely downtown in Sao Paulo, although it's a sprawling city. Moscow had a city airport - actually called Centralni - until about 2005, (opposite the Lenin Stadium on Leningradsky Prospect for those who know Moscow), but it's now closed and has been built upon.