How can one simulate a PDA with a FIFO queue PDA?
The answer is: "rotate".

Consider the stack like (bottom) ABCD (top). stack-pop is from top, queue-pop is from bottom (this is directed to myself, I am easily confused). Pushes are from top.

Now, if we want to stack-pop, we push a marker # (to top) then repeatedly queue-pop and push, essentially moving everything from bottom to top. If we delay the pushes by one, we are able to stack-pop, just not pushing the last symbol before the #. 

By the way, almost the same trick shows that we can even simulate the Turing machine by a single queue. Which was asked in our sister math department.