Let $A$ be a ring and $X$ be the spectrum of $A$ with the Zariski topology. For an element $f\in A$ let $X_f:=\{p\subset A\text{ prime ideal }\,|\,f\notin p\}$; the $X_f$ form a basis of the topology on $X$. Finally let $\mathcal{O}$ be the structure sheaf of $X$ (a sheaf or rings).
I have managed to show that the stalk of $\mathcal{O}$ at $\mathfrak{p}\in X$ is isomorphic to to the local ring $A_{\mathfrak{p}}$. What I'm struggling to understand though is the following:
Question 1: For $f\in A$ the ring $\mathcal{O}(X_f)$ is isomorphic to the localized ring $A_f$. Why?
Let $\mathfrak{N}$ be the nilradical of $A$. I know that the spectra of $A$ and $A/\mathfrak{N}$ are homeomorphic, but I have trouble answering this:
Question 2: Are the structure sheaves of the spectra of $A$ and $A/\mathfrak{N}$ the same?
I am very thankful for any hints, references or full out answers.