Build PDA for a language with unknown input alphabet
You might reason like this: If $L_1, L_2$ are regular, then $L_2^R$ is regular ($L^R$: reverse words). You can build regular grammars $G_1 = (N_1, \Sigma_1, P_1, S_1)$ and $G_2 = (N_2, \Sigma_2, P_2, S_2)$ that generate $L_1$ and $L_2^R$. The crucial point is that the regular grammars have the lonely non-terminal always at the end (or beginning!) of the sentential form. Build a CFG with nonterminals $Q = N_1 \times N_2$, and productions that build up $L_1$ to the left (using the first part of the nonterminal) and $L_2^R$ to the right (using the second part of the nonterminal, building from the end). From the resulting grammar you can build a PDA.
The basic idea is similar to the construction for $\{ w w^R \colon w \in \Sigma^*\}$ with $S \to x S x$ for all $x \in \Sigma$ and $S \to \varepsilon$ (or $S \to xx$ for purists).