Courant is one of the top places in the world to get a MS in Finance. This is what their website says:
Basically, you will need to have some grasp of Stochastic Calculus, Probability Theory, and Numerical Analysis. Some knowledge of Functional Analysis and Real Analysis/Measure Theory will help with the proofs but for the most part, from the point of view of writing Models, coding them up, simulating them, and calibrating them you will mostly be solving PDEs. You'll probably use MATLAB or C++ .
Your program may also have a strong econometrics basis, so you will need to develop data skills and learn how to use packages like SAS, STATA, LIMDEP etc. The theory will involve some linear algebra and probability theory.
The professors may not go very deep into Finance Theory but the type of math they use there involves Functional Analysis, Dynamic Programming, Martingales etc.
The program will be fast paced and challenging, especially Finance Theory since you won't have a base for it but you should be fine with the math part of it if you have an undergrad degree in math. Remember, you will understand things better with each iteration so try to get some working understand in your first pass and also try to have a bird's eye view of what's going on. It is easy to get lost in the details.