How should one approach this pde? Say I have $u=u(x,y)$ and I am facing $u_x=-yu_{xy}$. I tried to integrate both side wrt to $x$ but that was without any luck, any help is appreciated!
Try solving $v=-yv_y$ then solve $u_x=v$
When taking the partial derivative of a function, f(x,y), with respect to x, you treat y as a constant.  So when you integrate with respect to a single variable, you treat the other variable as a constant.  One caveat: the "constant of integration" might, in fact, be a function of that other variable.

Here you have $u_x= -yu_{xy}= -y(u_y)_x$.  Integrating with respect to x, $u= -yu_y+ c(x)$ where c(x) can be any differentiable function of x.  We can write that as $-yu_y+ u= c(x)$ and treat it as an ordinary differential in the single variable y with c(x) a constant (with respect to y).  That is a linear equation first order equation and if you are taking partial differential equations you should know how to find an "integrating factor" for a linear first order equation.