The following is basically a copy of dhy's comments with links (any mistakes are mine).

First, assume you know that for a finite surjective morphism $f:X\rightarrow Y$ between Noetherian integral affine schemes, there exists a positive integer $n$ and a morphism of $\mathcal{O}_Y$-modules $\mathcal{O}_Y^{\oplus n}\rightarrow f_*\mathcal{O}_X$ inducing an isomorphism at the generic point. This is shown in the linked question.

Now, we have a finite surjective morphism $f:X\rightarrow Y$ between Noetherian integral schemes. Pick a non-empty affine open $j:V\rightarrow Y$ whose inverse image is a non-empty affine open $i:U\rightarrow X$. The restriction $f|_U:U\rightarrow V$ is a finite surjective map (by definition). We know therefore that there is a positive integer $n$ and a morphism of $\mathcal{O}_Y$-modules $j_*\mathcal{O}_V^{\oplus n}\rightarrow f_* i_* \mathcal{O}_U$ inducing an isomorphism at the generic point. This can be composed with the natural map $\mathcal{O}_Y\rightarrow j_*\mathcal{O}_V$ to get a morphism of $\mathcal{O}_Y$-modules $\mathcal{O}_Y^{\oplus n}\rightarrow f_* i_* \mathcal{O}_U$ inducing an isomorphism at the generic point.

On any qcqs scheme any quasi-coherent sheaf is the directed colimit of its quasi-coherent subsheaves of finite type. Apply this to $i_*\mathcal{O}_U$. Since $X$ is locally Noetherian, these subsheaves are actually coherent.

We claim that the pushforward by a qcqs morphism, considered as a functor on the categories of quasi-coherent modules, commutes with directed colimits. First note that the functor of restriction to an open set commutes with colimits so the question is local on the target. Assume that the target is affine, then express the pushforward as a kernel. The conclusion follows from the fact that kernels commute with directed colimits and the fact that the pushforward by a morphism between affine schemes commutes with directed colimits.

Therefore, we can pushforward the the directed colimit to get a colimit of coherent sheaves on $Y$ (they will be coherent because $f$ is proper) coconverging to $f_*i_*\mathcal{O}_U$.

Finally, use Lemma 34 and Proposition 75 here (the word "concentrated" means qcqs) to conclude that any coherent sheaf on a Noetherian scheme is a finitely presented object of the category of quasi-coherent sheaves (i.e. $\mathrm{Hom}(F, -)$ preserves directed colimits). Therefore, the map $\mathcal{O}_Y^{\oplus n}\rightarrow f_* i_*\mathcal{O}_U$ factors through a coherent $\mathcal{O}_Y$-module $G\subset f_* i_*\mathcal{O}_U$. To see that $\mathcal{O}_Y^{\oplus n}\rightarrow G$ induces an isomorphism at the generic point, first notice that it must induce an injection at the generic point (because its composition with the inclusion is an isomorphism) and second notice that the generic rank of $G$ does not exceed that of $f_* i_*\mathcal{O}_U$. An injective map between finite-dimensional vector spaces such that the dimension of the target does not exceed the dimension of the source must be an isomorphism. Game over.