Typically not. The classic Japanese yumi was a compound longbow designed for horse archery. (The samurai may be known in modern times more for katanas, but the samurai were actually primarily horse archers that also happened to carry a melee weapon.) One cannot fire normal longbows (made for infantry) on horseback because the horse gets in the way, so the Japanese developed a highly asymmetric bow that is much longer on top, while most other horse archers used shortbows. Not to say they didn't have symmetric bows sometimes, particularly for ashigaru foot soldiers where they'd get cheaper bows that are easier to make, but the iconic Japanese samurai bow (and Akagi is going for the samurai look here) is asymmetric. Also, the yumi (bow) tag specifies only to use it for asymmetric bows.
Does having the bow longer on the top make for better shots at a downward angle?
Maybe if you're at an angle that's steep enough, but that's not the main reason. It's just that longbows are often as tall or taller than the person firing them, but it's impossible to properly draw a bow while seated if that bow goes further down than one's hip, since you need to draw across the body (which would be where the horse and saddle are when riding). (Similarly, you can't fire to the right when drawing with your right hand, just because you don't have room to pull, but horse archers would just turn their horse around so they could fire off to the left.)