Rio de Janeiro Sugar Loaf (which is actually a hill) and Corcovado Mountain (on which summit is located Christ statue) are not far from each other. In fact, from the roof top of many hotels you can see both.
The best and fastest way to go from Sugar Loaf to the bottom of Corcovado Mountain is by taxi. The taxi ride takes anything between 12 (in the morning on any day) and 25 minutes (or more, in the end of the afternoon on a weekday). In Rio, taxis are affordable and are the best and safest way for a tourist to get around.
Once you get off the taxi at Corcovado Train Station in Cosme Velho neighborhood, you buy your ticket to the next available train. The train ride up Corcovado Mountain takes 20 minutes. Once you get to the top of Corcovado you can stay as long as you want and enjoy the views of Christ the Redeemer statue and Rio de Janeiro.
If you want to learn how to avoid standing in line, save money on your ticket and much more, get a copy of Sugar Loaf: the definitive guidebook and Christ the Redeemer statue: a comprehensive guide.
See photos of both here: Sugar Loaf and Christ (plus practical travel information)