Let $G$ be a finite group and $S$ be a K3 surface. $G$ acts effectively and symplectically(fix the nowhere vanishing 2-form of $S$) on $S$. Since the action is symplectic,quotient surface $S/G$ has only rational double points. Let $S'$ be a desingularization of $S/G$.
Q1 How to show that $b_1(S')$(the first Betti number of $S'$)=0?
Q2 In general,blowing-up of one point (maybe singular point or nosingular point of complex surfaces $X$) $X'\to X$ preserves the first betti number? If $X$ has only rational double point ,this is true? Give me a reference please.
I'll first answer to the second question, and I assume you work over complex numbers. You only need to understand what happens for a single blow-up $X' \to X$ at $p \in X$ and can even look at the restriction to a small neighbourhood of $p$. For example, to understand the case when $X$ is smooth this is enough to see why the blow-up of $\Bbb C^2$ is simply connected.
In your situation, you need to look at double rational points, where this is also true because the exceptional divisor is simply connected (it's a graph of rational curves without cycles since it corresponds to ADE Dynkin diagrams) so you can use Mayer-Vietoris or you can by hand retract a ADE singularity on the exceptional divisor.
For the first question, this is true since a K3 is simply connected so again you can use Mayer-Vietoris.
I don't know if the second question holds in full generality, I guess not but can't think of a counter-example now.