Is $\langle (-y)/(x^2+y^2),x/(x^2+y^2) \rangle$ a conservative vector field? If it is,I found two potential function which is $\arctan(y/x)$ and $-\arctan(x/y)$.
But I don't know which is true.

Briefly: no, it's not a conservative vector field on its entire domain $\mathbb{R}^2 \setminus \{0\}$. The vectors of the vector field form counterclockwise circles about the origin, and the line integral of any counterclockwise circle about the origin is $2\pi$.
On any simply connected subset of its domain, however, the vector field is conservative. You can define a potential function as follows: pick an arbitrary point $p$ in the domain and let the potential at any point $x$ be the angle in radians from $p$ to $x$, where positive means you can get from $p$ to $x$ by moving counterclockwise within the domain and negative means moving clockwise. One possible choice of potential function is $\arctan(y/x)$, as long as the domain lies entirely in the first and fourth quadrants (otherwise there's a discontinuity associated with crossing the $y$-axis). Of course, it's not possible to define these potential functions unambiguously if the domain wraps all the way around the origin.
