No. This question is pretty closely related to this other question, but let me give an answer nonetheless:
Consider an elliptic curve $E$ in $\mathbb{P}^2(\mathbb{C})$, choose coordinates $[x:y:z]$ of $\mathbb{P}^2(\mathbb{C})$ in a way that the line $z = 0$ intersects $E$ at an inflection point $O$. Let $X := E \setminus \lbrace z = 0 \rbrace$. Choose a point $P \in E$ such that $P$ is not a torsion point with respect to the group structure on $E$ for which $O$ is the origin. Let $U := X \setminus \lbrace P \rbrace$. Then $U$ is an open affine subset of $X$, but there is no $f \in A:=\mathbb{C}[X]$ that vanishes only at $P$. So $\mathbb{C}[U]$ cannot contain $A_f$ for any non-constant $f \in A$.
