Christ the Redeemer sure has been the most iconic statue in Brazil, especially in Rio de Janeiro. as Rio's most iconic place, Christ the Redeemer statue is a must-visit place for tourist like you. Here are some facts you should know first of the Christ the Redeemer statue of Rio de Janeiro:

Christ the Redeemer (Portuguese: Cristo Redentor) is a 30 meters height Art Deco Statue created by French sculptor Paul Landowski and built by the Brazilian engineer Heitor da Silva Costa, in collaboration with the French engineer Albert Caquot. The face was created by the Romanian artist Gheorghe Leonida. (Wikipedia)

The statue is located at the peak of the 700-metre (2,300 ft) Corcovado mountain in the Tijuca Forest National Park overlooking the city of Rio. A symbol of Christianity across the world, the statue has also become a cultural icon of both Rio de Janeiro and Brazil, and is listed as one of the New Seven Wonders of the World. It is made of reinforced concrete and soapstone, and was constructed between 1922 and 1931. Christ the Redeemer is the 5th largest statue of Jesus. (Wikipedia)

How to get there? Some visitor/tourist have shared their experience in visiting this place in TripAdvisor, just like comments below:

If you can book a tour from your cruise ship - I'd consider that (they may have a city tour which includes other sites as well). Just verify how they get you to the the top of Cristo. If they make you take the train you are probably better off just taking a taxi on your own.
If taking a taxi just say "Corcovado" and specify the "Cosme Velho" train station (that is the where the base is for the train up). Ensure he turns on the meter. If you don't specify the Cosme Velho train station your cabbie may decide on his own to drive you 1/2 way up the mountain (where you then catch the train) and charge you an unmetered extra rate.
At the base there were literally 50 people holding signs and looking "semi-official" offering rides up in vans. I never inquired with these guys but wish I did.
At top, when we were waiting on a looong line to get back down - there was another line for National Park Vans to go down, but only if you came up with them. Not sure how to score those seats.
Sorry man, wish I had more specifics, but this is best I have, hopefully gets you in the right direction. (SamdogNY) Mar, 2013

Take a taxi from the port to Corcovado, take the train to the top, take a taxi back to the port.
Corcovado Information:
Hours: 8:30 a.m. to 7:00 p.m., the train leaving every half hour.
Ticket price for adults: R$46  (MonteCD) Feb, 2013

Or as explained by Hidden Pousadas Brazil below:
Option 1:
Go to the ticket office of the Corcovado train station in Cosme Velho or the Riotur kiosque on Copacabana beach, opposite Rua Hilário de Gouveia and buy your ticket there. First train up to Corcovado leaves 8.20 am.  Round trip cost R$46 adults; R$23 children.
The 583 bus will get you to Cosme Velho from  Copacabana; the 583, 570 or 584 from Ipanema or Leblon.

Option 2:
There is now  a van from Largo do Machado Square in front of the church. The cost is: R$45 high season and R$37 low season (despite what it says on the official site which is correct in poortuguese but not in English)
Payment can be made by cash or card and vans run 08.00 to 17.00 every day. The  last van down is between 18.00 and 18.30.
To get to Largo do Machado the easiest way is by metro/tube although there are lots of buses too. The metro stop has the same name as the square: ‘Largo do Machado’.

Option 3:
You can also take a van from half way up Corcovado mountain outside the former Hotel Paineiras (only way to get there is by car or taxi!). Cost R$27,36 in  high season; R$19,36 in low season.
Payment is cash only and vans run 08.00 to 17.00 every day; last van down is between 18.00 and 18.30

And of course, all the above tariff should have been updated as of 1 December 2015. However, you can still rely on the general process.