I would like to know, if its possible to plot the solution of the PDE:
Let $\lambda,p_1,p_2>0$ and $c_1, c_2 < 0$
I am looking now for some $f:(R^+_0)^2\rightarrow R^+_0$ such that
\begin{align*}
\lambda f(x,y)-\frac{1}{2}\triangle f(x,y)=p_1x+p_2y
\end{align*}
with the boundary conditions: $\frac{d}{dx}f(0,y)=c_1$ and $\frac{d}{dy}f(x,0)=c_2$
where $\triangle f:=\frac{d}{dx^2}f+\frac{d}{dy^2}f$
If its not to complicated to solve this problem with numerical Methods i would be verry glad if someone could tell me how to start
There is an issue with your problem formulation, before even considering numerics. You have a second order linear elliptic PDE in canonical form, which is a highly studied class, and typically is solved on a bounded domain with a single boundary condition on all boundaries. In some cases they can be solved on an unbounded domain with the far field limit specified.
Once you have found additional boundary conditions and limited your problem to a finite domain, you can construct a numerical version of your problem. The simplest approach if the domain is rectangular is to use a finite difference method, which should be acceptable in your case.
If you have access to a library with textbooks then introductory texts on elliptic partial differential equations and finite difference methods would be a good place to start.