The simple random walk on the circle graph $\mathbb{Z}/N\mathbb{Z}$ of size $N$ can be realized as the class modulo $N$ of simple random walk on $\mathbb{Z}$, hence
$$
p_{00}^{(n)}(\mathbb{Z}/N\mathbb{Z})=\sum_{k\in\mathbb{Z}}p_{0,kN}^{(n)}(\mathbb{Z}).
$$
Now, for every $k$,
$$
p_{0,k}^{(n)}(\mathbb{Z})=2^{-n}{n\choose (n+k)/2},
$$
with the convention that ${n\choose i}=0$ for every noninteger $i$ and every integer $i\le -1$ and every integer $i\ge n+1$.

On the other hand, this is also a random walk on a finite graph hence the stationary distribution allows to shortcut these exact computations for large $n$. For every odd $N$,
$$
\lim_{n\to+\infty}p_{00}^{(n)}(\mathbb{Z}/N\mathbb{Z})=\frac1N.
$$
For every even $N$, $p_{00}^{(2n+1)}(\mathbb{Z}/N\mathbb{Z})=0$ for every $n$ and
$$
\lim_{n\to+\infty}p_{00}^{(2n)}(\mathbb{Z}/N\mathbb{Z})=\frac2N.
$$