Equation of the circle passing through intersecting point of line and circle is given by $S+\lambda L=0$
where, $S x^{2}+y^{2}+y-1=0$
and $L x-y+1=0$
$\therefore $ Equation of the circle is
$\left(x^{2}+y^{2}+y-1\right)+\lambda(x-y+1)=0$
$\Rightarrow x^{2}+y^{2}+\lambda x+(1-\lambda) y+\lambda-1=0$
Centre of above circle $=\left(\frac{-\lambda}{2}, \frac{\lambda-1}{2}\right)$
Since, centre lies on $x-y+1=0$.
$\therefore \frac{-\lambda}{2}-\left(\frac{\lambda-1}{2}\right)+1=0 \Rightarrow \lambda=\frac{3}{2}$
Hence, required equation of circle is
$x^{2}+y^{2}+y-1+\frac{3}{2}(x-y+1)=0$
$\Rightarrow 2\left(x^{2}+y^{2}\right)+3 \,x-y+1=0$