Rio rewards even a single day if you move fast and stick to the icons. This itinerary skips Santa Teresa, the Botanical Garden, and the smaller museums on purpose, there simply isn't time, and focuses instead on the handful of sights that define the city.
Start early at Christ the Redeemer atop Corcovado, arriving right when it opens to beat both the crowds and the midday haze that can obscure the view. Come back down and head to Sugarloaf Mountain for the cable car ride and a second, lower-altitude panorama of the bay, ideally before midday heat and lines build up. For lunch, swing through Copacabana, grabbing a quick bite near the beach and a look at Copacabana Fort if time allows. In the early afternoon, walk the colorful Selarón Steps in Lapa, then pass under the nearby Arcos da Lapa on your way toward the historic center. Spend the mid-afternoon in downtown Rio, taking in the Metropolitan Cathedral, the Museum of Tomorrow on the revitalized waterfront, and a quick look at Cinelandia's grand square. As evening approaches, head to Ipanema for sunset over the water, then close the day with dinner along the beachfront, watching the lights come on over the coastline you've spent the day exploring.