Is there a way to generate a random variate from a non-standard distribution without computing CDF?
Indeed acceptance/rejection is the way to go if you know analytically your pdf. Let's call it f(x). Find a pdf g(x) such that there exist a constant c, such that c.g(x) > f(x), and such that you know how to simulate a variable with pdf g(x) - For example, as you work with a distribution with a finite support, a uniform will do: g(x) = 1/(size of your domain) over the domain.
Then draw a couple (G, U) such that G is simulated with pdf g(x), and U is uniform on [0, c.g(G)]. Then, if U < f(G), accept U as your variable. Otherwise draw again. The U you will finally accept will have f as a pdf.
Note that the constant c determines the efficiency of the method. The smaller c, the most efficient you will be - basically you will need on average c drawings to get the right variable. Better get a function g simple enough (don't forget you need to draw variables using g as a pdf) but will the smallest possible c.
I'm trying to write a Monte Carlo simulation. In my simulation I need to generate many random variates from a discrete probability distribution. 
I do have a closed-form solution for the distribution and it has finite support; however, it is not a standard distribution. I am aware that I could draw a uniform[0,1) random variate and compare it to the CDF get a random variate from my distribution, but the parameters in the distributions are always changing. Using this method is too slow. 
So I guess my question has two parts:

Is there a method/algorithm to quickly generate finite, discrete random variates without using the CDF?
Is there a Python module and/or a C++ library which already has this functionality?