Is it decidable that a context free language contains a given regular language?
Indeed, it is undecidable whether a context-free grammar generates all strings over its alphabet. But I guess you want to know how such a result is obtained.

The classical result "undecidable $L(G) = \Delta^*$ for CFG $G$" was obtained by Bar-Hillel, Perles and Shamir in 1961 using a coding of Turing Machine computations. They showed that the set of strings that where not codings of valid TM computations form a context-free language. Of course if there are no valid computations then that language is $\Delta^*$.

You can do the same with codings of solutions of the Post Correspondence Problem. Given set of pairs $(u_1,v_1), \dots, (u_n,v_n)$ you can code PCP "solutions" as strings $u_{i_1}\dots u_{i_k}\#v^R_{i_k}\dots v^R_{i_1}$ such that both sides are equal (or rather, mirror image).

Argue that the non-solutions form a context-free language.