I am able to verify the statements about the stalk. I want to see how the direct image of the the skyscraper sheaf can be thought of as the constant sheaf.
Observation- If $P\notin U$, then $U\cap {\{P\}}^{-}= \emptyset$ so the sections are just O which is same as the section of the skyscraper sheaf.
But if If $P\in U$ , I don't see why $i_{*}(A)(U)=A(U\cap\{P\}^{-})$ is equal to A .
Note that since the one-point set $\{P\}$ is irreducible, so too is its closure. And as Martin points out, any open subspace of an irreducible space is irreducible. I recommend doing Ex. I.1.6 if you haven't already.
I think the thing you want to prove in the end is that if $X$ is an irreducible topological space and $Y$ is a discrete space then any continuous map $f\colon X \to Y$ is constant. If you've gone through Chapter I then I think you know a quick proof of this already: by continuity, $f(X)$ has to be irreducible.