Let $f:X\to Y$ be a birational morphism. For a nonsingular variety $Z$, can any morphism $g:Z\to Y$ be lifted to a morphism $h:Z\to X$ so that $g=f\circ h$? Are there any related fact, except for the universal property of blow-up, that is known?
Precisely, I'm thinking of the following situation :
Let $C$ be a smooth complex projective curve and fix $p\in C$. Let $M_r$ (resp. $M_r^s$) be the moduli space of rank $r$, semi-stable (resp. stable) vector bundles $E$ over $C$ with $\det E=O_{C}$, and let $M_{r,(a_{1},a_{2})}^{par}$ (resp. $M_{r,(a_{1},a_{2})}^{par,s}$) be the moduli space of rank $r$, degree $0$, $(a_{1},a_{2})$-semi-stable (resp. stable) parabolic bundles $(E,H)$ over $C$, where $H\subset E|_{p}$ is a hyperplane and $(a_{1},a_{2})$ is a parabolic weight such that $0\le a_{1}<a_{2}<1$. We fix small $(a_{1},a_{2})$ so that $M_{r,(a_{1},a_{2})}^{par}=M_{r,(a_{1},a_{2})}^{par,s}$ and there exists a forgetful map $M_{r,(a_{1},a_{2})}^{par}\to M_r$ given by $(E,H)\mapsto E$.
Let $f:M_{2}\to M_{4}$ be a morphism given by $E\mapsto E\oplus E$. It is known that $f:M_{2}\to f(M_{2})$ is injective and birational such that $f:M_{2}^{s}\to f(M_{2}^{s})$ is an isomorphism. Consider the forgetful map $G:M_{4,(a_{1},a_{2})}^{par}\to M_4$, a nonsingular closed subvariety $S$ of $M_{4,(a_{1},a_{2})}^{par}$ and $g:=G|_{S}$. Here $S$ is the Zariski closure of $\{(F,H)\in M_{4,(a_{1},a_{2})}^{par}|\mathrm{End}(F)\cong M(2),\det F\cong O_{C}\}$ in $M_{4,(a_{1},a_{2})}^{par}$, where $M(2)$ is the matrix algebra of $2\times 2$ complex matrices. It is easy to see that $f(M_{2})=g(S)$ and $g:S\to f(M_{2})$. Because $S$ is nonsingular, we can lift $g:S\to f(M_{2})$ to $\tilde{g}:S\to M_{2}$. Why is it possible?
I assume $Y$ is singular. Let $h : Z \to Y$ be a resolution of singularity and $f = g \circ h$ where $g : X \to Z$ is the blow-up at a smooth point. The lift you are looking for will be the inverse of $g$ when restricted to the smooth locus of $Y$ and such inverse do not exist.
Related : the minimal resolution of an algebraic variety $X$ is the resolution $f : X \to Y$ with the property you mention. For surfaces, it exists and is given by any resolution $f : X \to Y$ so that $X$ do not contains $(-1)$ curves.