I hate it when the arena battles give me a bow. I am bad at bow.
BUT. I HAVE won 1v1s against a sword/shieldman on foot, and even a horseman whose loadout I do not recall. I felt damned awesome doing so.
As was said, you can can still get a guy with a shield in close combat if you shoot in the gap between where he lowers his shield and when he swings his weapon. I'm pretty sure you can nail 'em in the head or legs while they're approaching if you're a good shot or get lucky, too, but it's hard.
I don't remember the horseman fight very well, but IIRC it was also foggy, so it was sorta Phantom Ganon style where I was spinning around trying to spot him and get a shot off before he rode me down. I missed a lot, though, due to sucking at aiming, and a lot of the fight was dodging to the sides or trying to get run over instead of cut down. I probably could have backed up against a wall so he'd run into it and let me get a better shot, but I don't explicitly recall doing this.
I am also bad at couching and hitting on horseback, because I cannot the distance between my target and my horse. ...I'm just bad in general, I guess.
I haven't used a lance in a long time, actually, I think. I remember in earlier versions they'd do like 200-400 damage. Fun times.
On an completely digressive note, my favorite horse is the heavy steppe horse. Besides the fact that I usually never get to the point where I can outfit myself with higher up horses due to money or, more likely, riding skill, they also have the best maneuverability, IIRC, which is great. Heavy makes 'em less prone to getting cut out from under me, which is great.
Never really got to courser/warhorse tier, and I'm pretty sure I've only gotten plate armor through gold cheats, and at my best just use stuff looted off sea raiders or somethin'. I rarely buy anything, too - a lot of the time just food and occasionally horses.