1 [made Community Wiki]

It's rather subjective what is surprising or not. But there are lots of nontrivial and interesting equivalences of categories. Actually a big part of mathematics is about establishing these equivalences. Some examples:

1. The category of smooth projective curves over $k$ (alg. closed field) is anti-equivalent to the category of finitely generated field extensions of $k$ of transcendence degree $1$.

2. If $X$ is a nice pointed space with fundamental group $G$, then the category of coverings of $X$ is equivalent to the orbit category of $G$.

3. The category of locally compact hausdorff spaces is anti-equivalent to the category of commutative $C^*$-algebras. Similarily, replacing $\mathbb{C}$ by $\mathbb{F}_2$, the category of locally compact totally disconnected hausdorff spaces is anti-equivalent to the category of boolean rings. See also Johnstones book Stone duality for more background.

4. Tannaka duality, see for example this MO discussion.

5. Pontrjagin duality establishes an equivalence between the category of locally compact abelian groups and its opposite category. This restricts to an anti-equivalence between the category of compact abelian groups and discrete abelian groups!

Remark that in the first four examples, in some sense, geometry is connected with algebra. This not just enables you to argue about geometry with algebra, but also the other way round!

By the way, the equivalence between the category of finite sets and the non-negative integers is not surprising at all. Every category is equivalent to its skeleton, essentially by the definitions. A skeleton is some kind of classification of the objects, and a finite set is classified by its cardinality.