It is easy to find counterexamples with $X$ of general type. Take $S$ of general type with $\chi(S)=1$, $K^2_S>1$ and nonzero torsion in $Pic(X)$ (there are plenty of such surfaces, even with $h^1(\mathcal O)=h^1(\mathcal O)=0$). Now blow up $S$ to get $X$ and an effective $-1$ curve $E$ and set $D=E+L$, where $L$ is a non zero torsion element in $Pic(X)$. $D$ is a $-1$-class, but it is not effective: indeed if it were then it would have to contain $E$, since $DE=-1<0$, but then $L=D-E$ would be effective, contradicting the assumption that $L$ is non zero torsion.

ADDED: for rational surfaces one could try a direct approach, similar to the well known computation for finding $-1$-curves on a Del Pezzo surface of degree 3. $S$ is the blow up of $P^2$ or of a ruled surface $F_n$ at at most 8 (or 7) points. One knows the Picard group of $S$ and can try to write down explicitly the $-1$-classes and see whether they are all effective.