Karp's reduction strategy
$(3)\Rightarrow (1)$ should just be a consequence of the definition of $\textsf{NP}$-completeness.
For $(1)\Rightarrow (3)$, you can use the fact that there exists a function $f$, computable in polynomial time, such that $u\in X \Leftrightarrow f(u) \in Z$. Now to decide if $u\in X$, you can compute $f(u)$, and then decide nondeterministically if $f(u)\in Z$.