Context-free grammar to a pushdown automaton
This is basically the language of all palindromes over $\{a, b\}$ which can contain an arbitrary number of $b$'s in the middle. First, you could build a PDA for palindromes. This is a fairly standard language, so you should be able to find plenty to get you started. Then, you need to add a way to read an arbitrary number of $b$'s after you're done pushing new symbols to the stack, and before you start popping symbols. So the basic steps here:


Read $a$'s and $b$'s and push to the stack;
Read any number of $b$'s;
Read $a$'s and $b$'s and pop from the stack.


Please let me know if you need more guidance. Note that you're going to need to use nondeterminism here, so don't worry about it. This is a not a deterministic language (unless I'm mistaken... which has happened before).