I think that this might hold for any finite morphism $f: X \to Y$. We need to show that for a short exact sequence of sheaves on $X$, the sequence obtained by pushing forward $f_*(-)$ remains exact. I believe that this statement is syntomic local on the target by the definition of pushforward, so we may assume that $Y= Spec(R)$ is affine (and therefore so is $X = Spec(S)$).
I think that the proof of https://stacks.math.columbia.edu/tag/0531 shows that there is a finite syntomic cover $Y' = Spec(R') \to Y$ such that the base-change $f': X\times_{Y} Y' \to Y'$ factors as $f': X\times_{Y} Y' \xrightarrow{i} \bigsqcup_{j} Y' \xrightarrow{\pi} Y'$, where $i$ is a closed immersion and $\pi: \bigsqcup_{j} Y' \to Y'$ is the obvious morphism from the disjoint union of finitely many copies of $Y'$ (the syntomicity of $Y' \to Y$ is not explicitly stated, but follows from the construction). Therefore we are reduced to checking the exactness of $i_*$ and $\pi_*$. The OP already noted that the exactness for the closed immersion $i$ is shown in https://stacks.math.columbia.edu/tag/04C4. On the other hand, I think that the exactness of $\pi_*$ can be proven directly, as in that case pushing forward should be the same thing as taking a direct sum of sheaves (the restrictions to each copy of $Y'$ in the disjoint union), which is exact.