Quotient ring of a localization of a ring

Yes. This is because $A_p/IA_p \cong (A-p)^{-1}(A/I)$, and the latter ring is isomorphic to $(A/I)_{p/I}$ because the image of $A-p$ under the projection $A \rightarrow A/I$ is $A/I - p/I$.

One way to observe the first isomorphism is to note that $A_p/IA_p \cong A/I \otimes_A A_p \cong (A-p)^{-1}(A/I)$ (The latter isomorphism follows from the fact that given an $A$-module $M$, $S^{-1}A \otimes_A M \cong S^{-1}M$).

Later edit at OP's request: (To show the canonical map is an isomorphism) The canonical map $\overline{A}_{\overline{p}} \rightarrow A_p/IA_p$ is clearly surjective. Let $\frac{a+I}{s+I}$ be mapped to $0$. Then $\frac{a}{s} \in IA_p$. Then $ta \in I$ for some $t \in A-p$. So, $ta + I = (t+I)(a+I) = 0$, and from this you show that $\frac{a+I}{s+I} = 0$, proving injectivity.