I would like to know if it is possible to characterize the property "quasi-compact" in the category of schemes by means of a pure categorical language. This would imply in particular that every equivalence of categories $\text{Sch} \to \text{Sch}$ preserves quasi-compact schemes. Together with Jonathan's answer here, this would answer affirmatively my question about the rigidity of the category of schemes, at least over a field $k$.

For example the property of being empty is categorical, because it just says that the scheme is initial. The terminal scheme is $\text{Spec}(\mathbb{Z})$, so this is also categorical. Further examples of categorical properties or schemes: Spectra of fields, the underlying set of a scheme (in particular surjective morphisms), connected schemes, $\text{Spec}(\mathbb{Z}_p)$, and much more, see here. The usual definition of quasi-compact involves open immersions, which are, a priori, not categorical.