Here is a different way of computing the cohomology ring (which starts directly from the fan). The space is constructed by gluing to copies of $\mathbb{C}^2$ along an inclusion of $(\mathbb{C}^\times)^2$. The Mayer-Vietoris sequence associated to that would look like
$$
\cdots\to H^i(X_\Sigma)\to H^i(\operatorname{pt})\oplus H^i(\operatorname{pt})\to H^i((\mathbb{C}^\times)^2)\to H^{i+1}(X_\Sigma)\to\cdots
$$
and therefore induces isomorphisms $H^{i+1}(X_\Sigma)\cong H^i((\mathbb{C}^\times)^2)$ for $i>0$. What you get from the long exact sequence is a priori only additive information, but in this case there isn't much choice for the ring structure.

But there is more information: the pushout $X_\Sigma\cong\operatorname{colim}\left(\mathbb{C}^2\leftarrow(\mathbb{C}^\times)^2\rightarrow \mathbb{C}^2\right)$ is in fact a homotopy pushout. The homotopy type of $X_\Sigma$ can then be identified as $\Sigma(\mathbb{C}^\times)^2\simeq S^2\vee S^2\vee S^3$ because generally $\Sigma(X\times Y)\simeq \Sigma X\vee\Sigma X\vee \Sigma(X\wedge Y)$ - I apologize for the unfortunate notation but in this sentence $\Sigma$ is the suspension of topological spaces.
In this description, the ring structure is the one from $S^2\vee S^2\vee S^3$.

There is a motivic generalization of this. First of all, motives are supposed to explain the general patterns behind various sorts of cohomology theories in algebraic geometry. For smooth projective varieties, one has Chow motives and the motivic decompositions for smooth projective toric varieties are the reason behind the cycle class map being an isomorphism (as discussed in Fulton's book on intersection theory, see also the Chow ring formula in the toric varieties book). For smooth but not necessarily projective varieties, we will have to use Voevodsky's construction of the "derived category of mixed motives". If you are interested in studying smooth toric varieties and the relations between Chow rings and singular cohomology rings, I would strongly suggest to have a look at the motives literature - smooth toric varieties are linear, so their motives are mixed Tate and you get strong relations between Chow rings and singular cohomology (or $\ell$-adic cohomology and many others) from this.

Ok, so much for the general advertisement. The argument I gave above for computing the homotopy type of the variety works in what is called $\mathbb{A}^1$-homotopy theory. The covering of $X_\Sigma$ by two copies of affine space shows that the $\mathbb{A}^1$-homotopy type of $X_\Sigma$ is $\Sigma^{\mathbb{A}^1}\mathbb{G}_m^2\simeq\mathbb{P}^1\vee\mathbb{P}^1\vee\mathbb{A}^2\setminus\{0\}$. The motive $M(X_\Sigma)$ is then $\mathbb{Z}\oplus\mathbb{Z}(1)[2]\oplus\mathbb{Z}(1)[2]\oplus\mathbb{Z}(2)[3]$; the first three summands are what you see in Chow groups, the last summand can only be seen in Bloch's higher Chow groups. Applying the Hodge realization to the motive also gives you the Hodge structures, where the first three summands have the right weights (like they come from a smooth projective variety), but the last summand doesn't.

Finally, there are alternatives for doing the computations in motives. There is a localization (or Gysin) sequence for motives (or higher Chow groups) as well, so that the argument of abx can also be done in that setting.

I apologize that the motives part is maybe not very accessible if you have never seen motives before; I will see if I can give better explanations but full details would certainly be beyond the scope of this answer. Anyway, if you are interested in Chow groups of smooth toric varieties, I would say that motives are the way to go.