We certainly don't expect to get anything similar to the Sullivan conjecture. In the case where $G$ acts trivially, the Sullivan conjecture tells us that the space of maps from $B(\mathbb{Z}/p)$ to $X$ is homotopy equivalent to the space of constant maps, but that will not be the case if we replace $B(\mathbb{Z}/p)$ by the space $B(\mathbb{Z})=S^1$.

To explain how to compute what you do get, suppose that the generator of $\mathbb{Z}$ acts on $X$ as a homeomorphism $f:X\to X$. Using $\mathbb{R}$ as a model for $E\mathbb{Z}$, it is not hard to see that $X^{h\mathbb{Z}}$ is the homotopy pullback of the diagonal map $\delta:X\to X\times X$ and the map $\alpha:X\to X\times X$ given by $\alpha(x)=(x,f(x))$. At least if $X$ is simply connected and we take cohomology with coefficients in a field, this gives an Eilenberg-Moore spectral sequence
$$ \text{Tor}^{**}_{H^*(X)\otimes H^*(X)}(H^*(X),H^*(X))
\Longrightarrow H^*(X^{h\mathbb{Z}}). $$
Here the two copies of $H^*(X)$ in brackets are regarded as modules over $H^*(X)\otimes H^*(X)=H^*(X\times X)$ using $\delta^*$ and $\alpha^*$ respectively. I think that this is probably a practical method of calculation in some simple cases.