According to this paper: https://arxiv.org/pdf/math/0602562.pdf, in p.2 (below Theorem 1), it is written that if $S$ is a normal projective surface (so there are only finitely many isolated singularities) with quotient singularities (its definition is given in p.1) and $b_2(S)=1$, then either $\pm c_1(S)$ is ample or $c_1(S)=0$, but I can't see why. (I think the condition $b_2(S)=1$ should be critical.) Can this be proved using the following argument? Or is there another way to prove this?
Recall that $c_1(S)$ is ample if and only if $c_1(S)^2>0$ and $c_1(S)\cdot C>0$ for every irreducible curve $C$. Suppose $c_1(S)$ is nonzero. Since $b_2(S)=1$ we have $H^2(S;\Bbb Z)=\Bbb Z$ (we ignore torsion). Let $\alpha$ be a generator of $H^2(S;\Bbb Z)=\Bbb Z$, then $c_1(S)=k\alpha$ for some nonzero $k$. Also (the homology class of) every curve is of the form $m\alpha$ for some $m\in \Bbb Z$ (we identify $\alpha$ with $PD(\alpha))$. We consider the cases $\alpha^2>0$ and $\alpha^2=0$ and $\alpha^2<0$.
Yeah, you are on the right track. Since the variety has quotient singularities is an orbifold, also it may be seen that the restriction of the Kahler form $\omega$ is a orbifold differential form in the sense of Satake.
Then following this orbifold set up it is straightforward to prove that $[\omega] \in H^{2}(M,\mathbb{R})$ is well defined and it will has familiar properties. i.e. $\int_{S} [\omega]^2>0$ and $[\omega].C >0 $ for any curve $C$, since $C$ is complex and hence Kahler.
This was proven by Satake in the original paper on orbifolds from the 1950's, where he proved that De Rham cohomology works in the orbifold setting.
Then we may write $C_{1}(S) = k [\omega]$ and we get the 3 cases depending on whether $k>0$(Fano) $k=0$ (generalised Calabi Yau) $k<0$ (general type).
This fact is true in higher generality (for example for $\mathbb{Q}$-Gorenstein varieties), with essentially the same argument. But defining the relevant objects requires developing intersection theory on singular varieties, see for example Fulton or Hartshornes book.