Every ring homomorphism to the integers is surjective?

The image of a ring homomorphism (preserving $1$) is a subring of the codomain.

Since the only subring of $\mathbb{Z}$ is $\mathbb{Z}$ itself, the statement is proved.

The same is true for ring homomorphisms to $\mathbb{Z}/n\mathbb{Z}$.

Your proof is good as well, but overkill.