Let me do some checking.
Be back to you with something firm in a day or two.
In “Western”, ie, Greek, battle formations, bowmen were looked down upon. But in Middle Eastern (desert) armies, they were a very practical and effective way to fight. And, yes, the fact that the great Jehu was a skilled archer tells us quite a bit about how an archer would have been regarded in the Israelite command structure. I get the impression anyone trained to command would have been expected to be a good archer.
I’ll go a step further with that thought. On all the stele and hiroglyph walls, the kings brag about their prowess at hunting lions and gazelle and their enemies with either bows or spears. You see it in the images and the friezes. So I think you’re completely safe to pick one or both of those as Ahab’s weapons of choice.
The most useful civilian thing any man could do would be to protect flocks and crops from marauders. The best ways of doing that would be either with bows or slings. And I just have the impression from David’s story that a sling was considered something of a more proletarian weapon.