= Solution
Let $D$ be a <valuation ring>, $K$ its <fraction field>, and let $u:\operatorname{Spec}K\to X$, $v:\operatorname{Spec}D\to S$ be a compatible pair. Since $D$ is a <local ring>, every <open subset> of $\operatorname{Spec}D$ containing its unique <closed point> is the whole space: a <principal open subset> containing the <maximal ideal> is defined by a <unit>. Choose an <affine open subscheme> $V=\operatorname{Spec}A\subseteq S$ containing the image under $v$ of that <closed point>. The entire $v$ factors through $V$.
Because $f$ is a <finite morphism>, $f^{-1}(V)=\operatorname{Spec}B$ with $B$ a finite $A$-<module>. The valuative problem is the compatible <ring homomorphisms>
$$
A\longrightarrow D\subseteq K,\qquad \psi:B\longrightarrow K.
$$
Every $b\in B$ is an <integral element> over $A$. Mapping its <monic polynomial> equation to $K$ proves that $\psi(b)$ is an <integral element> over $D$. Part (c) gives $\psi(b)\in D$. Thus $\psi$ factors uniquely as a <ring homomorphism> $B\to D\hookrightarrow K$, producing the unique extension $\operatorname{Spec}D\to X$.
For completeness, a <finite morphism> is a <quasi-compact morphism> and of <finite type>: over $V$ its inverse image is an <affine scheme>, and finite $A$-<module> generators also generate it as an $A$-<algebra>. It is also a <separated morphism>. Affine-locally its <diagonal morphism> corresponds to the surjective multiplication <ring homomorphism>
$$
B\otimes_AB\longrightarrow B,\qquad b\otimes c\longmapsto bc,
$$
and hence is a <closed immersion>. In particular it is a <quasi-separated morphism>. The <valuative criterion for properness> now applies, giving
$$
\boxed{f\text{ finite}\Longrightarrow f\text{ proper}.}
$$
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