"Cohomologically minimal" resolution of singularities Let $X$ be a projective variety over $\mathbb{C}$. 
Is there a resolution of singularities $\tilde{X} \rightarrow X$ such that 
$H^*(X,\mathbb{Q})\rightarrow H^*(\tilde{X},\mathbb{Q})$
is surjective?
 A: No. Consider $X=\{x_0^2+x_1^2+x_2^2=0\}\subset \mathbb P^3$, the quadric surface with a node, which can be viewed as a cone over a conic $C$ and has an ordinary at the cone point. The minimal resolution $\tilde{X}$ of $X$ is the blowup at the nodes, but $\tilde{X}$ is $\mathbb P^1$-bundle over $C$ (actually the Hirzebruch surface $F_2$). Since $H^2(X,\mathbb Z)=\mathbb Z$, while $H^2(\tilde{X},\mathbb Z)=\mathbb Z\oplus \mathbb Z$, this cannot be a surjective.
In general the minimal resolution of an ADE surface singularity will be a counterexample.
Now I claim surjective is not possible for all resolutions of $X$. Since any resolution $X'\to X$ will factor through the minimal resolution $\tilde{X}\to X$, there is a surjective map between smooth projective varieties $X'\to \tilde{X}$, which induces a cohomology inclusion $$H^2(\tilde{X},\mathbb Q)\hookrightarrow H^2(X',\mathbb Q).$$ Therefore $H^2(X',\mathbb Q)$ is at least rank two. Here I used a following lemma from Voisin's Hodge theory, vol I, Lemma 7.28:

Lemma: Let $\phi:Y\to Z$ be a surjective holomorphic map between two compact complex manifold with $Y$ Kahler. Then $\phi^*:H^k(Z,\mathbb Q)\to H^k(Y,\mathbb Q)$ is injective for all $k$.

