Let $M$ be a an uncountable monoid (like $\mathbb R$ with addition or multiplication) and $N$ be a countable monoid (like $\mathbb N_0$, or $\mathbb Z$ with addition or multiplication). Further suppose we have a surjective homomorphism $\varphi : M \to N$ (like $(\mathbb R, \mbox{max})$ and $\mathbb (\mathbb Z, \mbox{max})$ with $\varphi(x) := \mbox{min}\{ k \in \mathbb Z : x \le k \}$, i.e. rounding up).
In this situation, does there always exists a homomorphism $\psi : N \to M$ such that $\varphi \circ \psi = \mbox{id}_N$.
For the terminology as used in the title, see wikipedia:surjections as epimorphisms.
No. We can even write down a counterexample for abelian groups: take $M = \mathbb{R} \oplus \mathbb{Z}$ and take $N = \mathbb{Z}/2\mathbb{Z}$. If you really want $N$ to be countable as opposed to at most countable, take $N = \mathbb{Q} \oplus \mathbb{Z}/2\mathbb{Z}$ (there's a surjection $\mathbb{R} \to \mathbb{Q}$ assuming the axiom of choice; without choice replace $\mathbb{R}$ with $\mathbb{Q}^{\mathbb{N}}$).
More generally, in the setting of groups at least, look up the group extension problem.