at.algebraic topology - Connectivity after Geometric Realization?

Yes it is a surjection on $pi_0$, because each component of $|Y|$ has at least one component of $Y_0$.

Beyond that there are no restrictions. For instance, you can get any homotopy type for $|X|$ and $|Y|$ and any homotopy type for the map between them with $X_0$ and $Y_0$ just one point, as long as you ask that $pi_0(|X|)$ and $pi_0(|Y|)$ are trivial.

(I'm taking "simplicial space" to mean a simplicial object in the category of topological spaces, say the compact Hausdorff ones.)