As $AG$ is noetherian of dimension 1*, finitely-generated projective $AG$-modules are classified by their rank and determinant, and so
$$K_0(AG) = \mathbb{Z} \oplus Pic(AG).$$
Proposition 10.3.8 of tom Dieck--Petrie gives an exact sequence calculating $Pic(AG)$ in terms of the ghost map, and for $G=C_p$ cyclic of odd prime order this is easy to work out their sequence manually and get
$$Pic(AC_p) \cong (\mathbb{Z}/p)^\times/\{\pm 1\} \cong \mathbb{Z}/((p-1)/2).$$

\* Certainly $AG$ is finitely generated, so noetherian. As the ghost map $\Phi : AG \to CG \cong \mathbb{Z}^{\times N}$ is an injective ring homomorphism, and $\vert G \vert \cdot CG \subset \Phi(AG)$, it follows that $CG$ is a finite $AG$-module. But $CG$ is a product of 1-dimensional rings, so is 1-dimensional, and hence $AG$ is also 1-dimensional.