Let $k$ be a field, let $A$ and $B$ be commutative, finitely-generated, graded $k$-algebras, and let $S$ be a finitely-generated, graded $k$-subalgebra of $B$.
Question1: If $\varphi:A\to B$ is a graded $k$-algebra map, must $\varphi^{-1}(S)$ be finitely generated?
Question2: If so, can we say anything about the degrees of generators for $\varphi^{-1}(S)$? More specifically, if $A$, $B$, and $S$ are generated in degrees at most $d_A$, $d_B$, and $d_S$, is there some constant $C=C(d_A,d_B,d_S)$ for which there must exist generators of $\varphi^{-1}(S)$ in degrees at most $C$?
This paper shows that in characteristic $0$, the intersection of two finitely-generated subalgebras need not be finitely generated, so that $\varphi(A)\cap S$ may not be finitely generated. Then, since $\varphi^{-1}(S)=\varphi^{-1}(\varphi(A)\cap S)$, we have a counterexample to question $1$. In Remark-Question 1.6, the author states that the results of the paper do not extend to positive characteristic.
Does characteristic $p$ help? Or are there other properties of $A$, $B$, $S$, and/or $\varphi$ for which my questions have affirmative answers? Sorry, for the vagueness here - I wish I could be more specific than "other properties," but this seems like a hard question and I'm not quite sure what should be true of my setup to make it work. I'd be happy with partial answers and/or references. Thanks.