= Solution
Use the <Minkowski metric> $g^{\mu\nu}=\operatorname{diag}(1,-1,-1,-1)$ throughout. Taking the <Hermitian conjugation> of the massive <Dirac equation> and using the <Dirac adjoint> gives
$$
i(\partial_\mu\bar\psi)\gamma^\mu+m\bar\psi=0.
$$
Taking the <matrix transpose> and multiplying by the <charge conjugation> matrix $C$ yields
$$
iC\gamma^{\mu T}C^{-1}\partial_\mu(C\bar\psi^T)+mC\bar\psi^T=0,
\qquad
-i\gamma^\mu\partial_\mu\psi^c+m\psi^c=0.
$$
Thus the <charge conjugation of a Dirac field> preserves the free massive <Dirac equation>:
$$
\boxed{(i\gamma^\mu\partial_\mu-m)\psi^c=0.}
$$
The <unitarity> of $C$ and $\gamma^0C^{-1}\gamma^0=-C^{-1}\gamma^{0T}$ also give
$$
\overline{\psi^c}=(C\bar\psi^T)^\dagger\gamma^0
=\psi^T\gamma^{0T}C^{-1}\gamma^0=-\psi^TC^{-1}.
$$
Consequently
$$
\boxed{\hat C\bar\psi(x)\hat C^{-1}=-\psi^T(x)C^{-1}.}
$$
Here $\hat C$ is the <unitary operator> acting on the <relativistic quantum field>, whereas $C$ acts on <Dirac spinor> indices.
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