Yes, you can rotate the bars backward and bring the hoods up, and it's probably a good idea. The setup now is pretty old school, in the sense that the sloping ramps that lead to the low lever positioning is a bar design from before the era (which we're in right now, and your levers are, too) when using the lever hoods was a primary grip. The bar may even be a track bar, where no levers are used at all and the primary handhold is the drops.

So yeah, you can rotate the bars back, but be mindful that using the drops will limit how far you can go because if they're too angled, and there is no bar under your hand, keeping your hands on the bar will be harder and the risk of a bump knocking your hands right off will be higher.

The other thing to consider is a bar shape better suited to your needs. Looking at your stem, it looks like an old 25.4 clamp diameter, which limits things a bit, but check out Soma Fab, for example, who offer a great selection, including the compact shape HWY1, in that diameter.