I don't have Arnold, but I can get it working with physical, which may help you regardless of renderer...
But without it, I can't come up with a solution without making the hair physical, but if you choose circle or similar type in the generator tab of the hair object you can force it to create hair geo (and can turn hair render off), which you can then texture using regular projections, which should stick to the hair geo, like so.
There is no reason to think that wouldn't also work in Arnold, though I wouldn't be surprised if you had to actually make the hair editable, and bake down a cubic or flat projection by deleting any existing UV map, selecting the projecting material, and doing 'generate UV cords' which would then the projection into the mesh. So, perhaps a fall-back method if there isn't a forthcoming answer directly within Arnold hair / material settings...
CBR