Three days in Rio de Janeiro.
Three days is enough to see the two great icons, spend real time on the beach, and get a feel for Rio's older, hillier heart — as long as you don't try to cram all of it into one morning. This is the pace we'd suggest to a first-time visitor: one big sight a day, anchored around the beach the trip is really for.
It assumes you're based in the South Zone, which keeps everything close. Swap the order to follow the clearest weather — you want a clear day for Christ the Redeemer and Sugarloaf, so watch the forecast and be flexible.