This statement isn't true in all abelian categories. The morphism $\bigoplus \mathbb{Z} \to \prod \mathbb{Z}$ from an infinite coproduct to an infinite product of copies of $\mathbb{Z}$ is monic but not epic in $\text{Ab}$, so the corresponding morphism in $\text{Ab}^{op}$ is epic but not monic. Freyd-Mitchell doesn't help here because the embedding it gives is only guaranteed to be exact; it isn't guaranteed to preserve infinite products or coproducts.

(Note that by Pontrjagin duality $\text{Ab}^{op}$ is the category of compact Hausdorff abelian groups, with $\mathbb{Z}$ being sent to $S^1$, so the claim is that the canonical morphism from an infinite coproduct to an infinite product of copies of $S^1$ is epic but not monic. This might seem strange, but the infinite coproduct in compact Hausdorff abelian groups is not the infinite direct sum: it's the Bohr compactification of the infinite direct sum.)

This statement isn't true in all abelian categories. The morphism $\bigoplus \mathbb{Z} \to \prod \mathbb{Z}$ from an infinite coproduct to an infinite product of copies of $\mathbb{Z}$ is monic but not epic in $\text{Ab}$, so the corresponding morphism in $\text{Ab}^{op}$ is epic but not monic. Freyd-Mitchell doesn't help here because the embedding it gives is only guaranteed to be exact; it isn't guaranteed to preserve infinite products or coproducts.

(Note that by Pontrjagin duality $\text{Ab}^{op}$ is the category of compact Hausdorff abelian groups, with $\mathbb{Z}$ being sent to $S^1$, so the claim is that the canonical morphism from an infinite coproduct to an infinite product of copies of $S^1$ is epic but not monic. This might seem strange, but the infinite coproduct in compact Hausdorff abelian groups is not the infinite direct sum: it's the Bohr compactification of the infinite direct sum. Also note that $\text{Ab}$ is locally finitely presentable and $\text{Ab}^{op}$ isn't, so maybe that's a hint as to what the correct conditions to impose are?)