You get a very good chance to hit your direct manager in the back (Because of a very big mistake he has done). What would you do?

Well , this is not the way , but correcting his mistake is my debate because I may face the same problem in future

I will try to correct the mistake if I can, otherwise I will wait for the management decision.

You are working in a company not in a jungle. The first and most important point is to work as a team and then provide as much support as possible. Mistakes are possible by anyone and realizing the mistake and rectifying them within the table and immediately is the best solution. Hitting at the back of your direct manager will definately get you the position you've dreamed of but at the same time you will loose respect from your juniors and the work environment will be more like a pain than enjoyable.

 If I was in this situation I would highlight and address this situation directly with the manager and work with them to improve and correct the mistake. Afterall a mistake made by one would reflect on the entire department, and hence collective work and coordination is very important. There should never be an insticnt to hit anyone in the back be it a subordinate or a manager. A good and healthy work environment can only be built when people work together with trust.

You diffenately dont need to do that.

I did encounter that situation few times at work. One time he was out of the office due to personal reasons (without letting know about that fact the board of directors) and received in the same time a task with dead line in the same day and the second time he forgot about one of his tasks and notice it only1 day before the dead line. I completed those task for him, forward it to his email the works and he sent it to the upper management.

First of all, we are team mates, we are paid with salaries for our work if our company deliver quality work in time for clients. Second, I didn't wanted his position. If I wanted it, I would apply for it. Third, there is no use to identify/reveal a problem if you dont have the solution ready. Lastly, if I were member of the board of directors I would not promote an employee as a result of his "assassination" skills. Maybe only when assassination is the company business.