Archery was quite popular among the European nobility, but in almost all cases, this was because of hunting, not war. Whether done on horseback with huge packs of dogs, or on foot with a bow, hunting was huge. It's difficult to overstate just how important a role it played in the lives of European aristocrats; any other hobby was decidedly of secondary priority. Some will tell you that this was because hunting was a form of practice for war, but while hunting teaches and reinforces many warlike traits (the ability to ride, to cooperate with a group of other horsemen in pursuit of a common objective), it's not the entire answer. Aristocrats hunted because they enjoyed it, full stop.

As far as examples of aristocrats who we know could string a bow? William the Conqueror has come down to us as a great fan of archery. His firstborn son, William the Red, died in an archery "accident" while hunting in the New Forest.

But no, horse archery was almost unknown in western European warfare. There are some examples of it in the historical record, but they are rare and basically all from the 15th century. Knights didn't only fight on horseback, but the style of war they practiced was based around shock, not fire. If mounted, they fought with lance and sword; if dismounted, they fought with lance and sword (and some other things, but cut-down lances and swords were the most common).