This is true in a much more general setting:

Fact Let $X$ be a proper variety over a noetherian ring $A$ and let $\mathscr L$ be a line bundle on $X$. Then $\mathscr L$ is ample if and only if $\mathscr L_{\mathrm{red}}\simeq \mathscr L\otimes \mathscr O_{X_{\mathrm{red}}}\ $ is ample on $X_{\mathrm{red}}$. (See Exc.5.7(b) in [Hartshorne]).

In your case $X_0\simeq X_{\mathrm{red}}$ and $\omega_{X_0}\simeq \left(\omega_{X} \right)_{\mathrm{red}}\simeq \left(\omega_{X/\mathbb C[\varepsilon]}\ \ \right)_{\mathrm{red}}$ .

This is true in a much more general setting:

Fact Let $X$ be a proper variety over a noetherian ring $A$ and let $\mathscr L$ be a line bundle on $X$. Then $\mathscr L$ is ample if and only if $\mathscr L_{\mathrm{red}}\simeq \mathscr L\otimes \mathscr O_{X_{\mathrm{red}}}\ $ is ample on $X_{\mathrm{red}}$.

See Exc.III.5.7(b) in [Hartshorne] for the proper case and EGA II, 4.5.13 without the proper assumption (thanks to @Laurent Moret-Bailly for the EGA reference).

In your case $X_0\simeq X_{\mathrm{red}}$ and $\omega_{X_0}\simeq \left(\omega_{X} \right)_{\mathrm{red}}\simeq \left(\omega_{X/\mathbb C[\varepsilon]}\ \ \right)_{\mathrm{red}}$ .