I have a strong feeling that, for a compact connected Riemann surface $X$ of genus $g>0$, the Euler characteristic of the Weierstrass divisor $W$ equals $$\chi(X,\mathcal{O}_X(W)) = (g-1)^2.$$ Is this true?

Answer:

By Riemann-Roch, the Euler characteristic is given by $$ \chi(X,W) = g^3 -g + 1- g = g^3-2g+1.$$

This is not equal to $(g-1)^2$ for $g>1$.