I need help with Showing that every finite automaton is a one-state pushdown automaton
The best place for such questions is the Discrete Math subforum.
Strictly speaking, a finite automaton is not a pushdown automaton because it does not have a stack. You probably need to emulate a finite automaton with a one-state pushdown automaton. What do you think about having a single symbol in the stack that represents the state of the finite automaton?