$\exists$ a non-compact space in which every proper open subset is compact?
Consider $\Bbb N$, with proper open sets given by $U_n = \{x: x\le n\}$ and the empty set. Arbitrary unions of the $U_n$ are open, (either given by $U_{m}$ the maximum of the $n$ or by $\Bbb N$ if the $n$ are unbounded), as too are finite intersections.
Here every proper open set is finite, and thus trivially compact. However, it is easy to see that $\Bbb N$ itself is not compact with this topology, since it is covered by the collection of all proper open subsets, which admits no finite subcover.