PDA for this context-free grammar
From CFG to PDA is a simple construction. Set axiom $S$ as initial pushdown symbol. If the topmost symbol of the stack is nonterminal replace it according to a production of the grammar. If the topmost symbol is a terminal we are forced to read the same symbol from input tape (and pop the symbol). That is called expand-match. See wikipedia. The resulting PDA is single state, acceptance by empty stack.

(added) Your problem is on a special type of grammar, linear context-free. That means in every step of the derivation there is exactly one nonterminal. You can mimic this using a different construction. Basically the nonterminal is the state. All terminal symbols to the left are read, those to the right are pushed, to be read later. For $S\to aaBb$ the PDA goes from state $S$ to state $B$ while reading $aa$ (in usual models that takes two steps) and pushing $b$. 

On your construction (it seems of the latter type). Concretely: the grammar generates $a$'s and $b$'s in a two-to-one / one-to-two ratio. In general: in order (for yourself) to see why it doesn't work you have to try to formulate the meaning of the states. How do they relate to things happening in the grammar?