I'll do it step by step. First lower all the indices:
\begin{equation}
-\frac{1}{4}F_{\mu \nu}F^{\mu\nu}=-\frac{1}{4}F_{\mu \nu}g^{\mu\rho}g^{\nu\sigma}F_{\rho\sigma}
\end{equation}
then expand the products,
\begin{equation}
-\frac{1}{4}(\partial_{\mu}A_{\nu}\partial_{\rho}A_{\sigma}-\partial_{\mu}A_{\nu}\partial_{\sigma}A_{\rho}-\partial_{\nu}A_{\mu}\partial_{\rho}A_{\sigma}+\partial_{\nu}A_{\mu}\partial_{\sigma}A_{\rho})g^{\mu\rho}g^{\nu\sigma}
\end{equation}
Using the symmetry of the metric, I can rename the indices and then switch them, to get
\begin{equation}
-\frac{1}{4}(2\partial_{\mu}A_{\nu}\partial_{\rho}A_{\sigma}-2\partial_{\mu}A_{\nu}\partial_{\sigma}A_{\rho})g^{\mu\rho}g^{\nu\sigma}
\end{equation}
Now let us take a functional derivative with respect to $\partial_{\lambda}A_{\alpha}$
\begin{equation}
\frac{\partial}{\partial(\partial_{\lambda}A_{\alpha})}(\partial_{\mu}A_{\nu}\partial_{\rho}A_{\sigma})g^{\mu\rho}g^{\nu\sigma}=(\delta^{\lambda}_{\mu}\delta^{\alpha}_{\nu}\partial{\rho}A_{\sigma}+\delta^{\lambda}_{\rho}\delta^{\alpha}_{\sigma}\partial_{\mu}A_{\nu})g^{\mu\rho}g^{\nu\sigma}=2\partial^{\lambda}A^{\alpha}
\end{equation}
Similarly for the other term. Hence,
\begin{equation}
\frac{\partial}{\partial(\partial_{\lambda}A_{\alpha})}\left(-\frac{1}{4}F_{\mu \nu}g^{\mu\rho}g^{\nu\sigma}F_{\rho\sigma}\right)=-\left(\partial^{\lambda}A^{\alpha}-\partial^{\alpha}A^{\lambda}\right)=-F^{\lambda\alpha}
\end{equation}
Now setting $\lambda=0$ gives the desired result.
