At least for a projective variety over a field you can use Seshadri's criterion.
Let $f:X\to Y$ be finite and $D\subset Y$ an ample (Cartier) divisor. By Seshadri's criterion there exists an $\varepsilon>0$, such that for any proper curve $B\subseteq Y$, and any $y\in B$, $$ \frac{D\cdot B}{\mathrm{mult}_y B}\geq \varepsilon.$$
Now consider any proper curve $C\subseteq X$ and any $x\in X$. Let $B=f_*C$ and $y=f(x)$. Since $f$ is finite, $B$ is an actual curve. (I.e., if it weren't finite, $B$ could be zero as a $1$-cycle.) Then since $f^*D\cdot C=D\cdot f_*C$ and ${\mathrm{mult}_xC}\leq {\mathrm{mult}_yB}$, we have $$ \frac{f^*D\cdot C}{\mathrm{mult}_xC}\geq \frac{D\cdot B}{\mathrm{mult}_y B}\geq \varepsilon,$$ which implies that $f^*D$ is ample by Seshadri's criterion again.
Remark I think one can prove the same using the definition of ampleness by making coherent sheaves globally generated, but I did not know whether that would count as using cohomology.
