Many dualities between geometry and algebra arise via a dualizing object. Roughly, if $\mathcal C$ is a category of spaces and $\mathcal D$ a category of "algebras", one often finds a dualizing object $R$ which lives in both categories $\mathcal C$ and $\mathcal D$ such that the constructions $C\mapsto \hom_\mathcal C(C,R)$ and $D\mapsto \hom_\mathcal D(D,R)$ constitute an equivalence or at least adjunction between $\mathcal C$ and $\mathcal D$.
For instance, Pontrjagin duality ($R=\mathbb R/\mathbb Z$), Stone duality ($R=\mathbb Z/(2)$), Gelfand duality ($R = \mathbb C$), and the fundamental theorem of Galois theory ($R=\bar k$, for $k$ a field) arise in this way.
Question: What is the dualizing object in the duality between affine schemes and commutative rings?
(Second question: What is the dualizing object in the fundamental theorem of covering spaces, which roughly states that the category of covering spaces over a space $X$ is equivalent to the category of all sets equipped with an action of the fundamental groupoid on that set? On the one hand, it is not a duality, so maybe it doesn't have a dualizing object in the strict sense, but since this statement is very similar to the fundamental theorem of Galois theory, maybe there's something similar.)
Your question presumes that a dualising object exists. My view is that it does not exist. On the affine scheme side, the affine line $\mathbb{A}^1$ has a ring structure and represents the $\textbf{Aff}^\textrm{op} \to \textbf{CRing}$ half of the equivalence – this is unproblematic. But in the philosophy of the dualising object conception of duality, there would have to be a ring $R$ that "is" in some sense also $\mathbb{A}^1$ (in a "covariant" way!) that "represents" the $\textbf{CRing}^\textrm{op} \to \textbf{Aff}$ half of the equivalence and I do not see any reasonable way of interpreting this precisely.
The situation is a lot better if we instead look at affine varieties over an algebraically closed field $k$ and integral domains finitely generated over $k$. The dualising object is then $\mathbb{A}^1_k$ as a variety and $k$ as an algebra. This is "obviously correct": as before, $\mathbb{A}^1_k$ has a ring structure, and moreover the set of points of $\mathbb{A}^1_k$ is canonically identified with the set of elements of $k$, so there is a good sense in which we can think of $\mathbb{A}^1_k$ and $k$ as being "the same". Furthermore, $k$ does in fact represent the functor that sends an integral domain finitely generated over $k$ to the set of points of the affine variety it corresponds to.
Given the above, it would seem that the ring incarnation of $\mathbb{A}^1$ should be $\mathbb{Z}$ – but, frankly, this is unconvincing. While it is true that we can canonically identify the elements of $\mathbb{Z}$ with certain points of $\mathbb{A}^1$, there are many more points besides. It is also difficult to say that the functor represented by $\mathbb{Z}$ is a functor that sends a ring to the set of points of the scheme it corresponds to – at least, it certainly does not represent the functor sending a ring $A$ to the set of prime ideals of $A$. For these (and other) reasons, I think it is not reasonable to say that there is a dualising object in this story.