Let $f:A\rightarrow B$ be a map of integral domains making $B$ into a finitely generated $A$-algebra. Suppose that $f^{-1}$ induces a bijection from the set of prime ideals of $B$ to the set of prime ideals of $A$. Is $B$ a finitely presented $A$-algebra?
In the case $A$ is Noetherian it is true (because any finitely generated $A$-algebra is finitely presented).
If $f^{-1}$ is only required to induce an injection on prime ideals, take a non-Noetherian integral domain with a prime ideal that is not finitely generated; the map $A\rightarrow A/I$ is a counterexample.
If $f^{-1}$ is only required to induce a surjection on prime ideals, take the ring of polynomials in infinitely many variables over an algebraically closed field; the map $A\rightarrow A[x]$ is a counterexample.