I can answer this question in regards to Feudal Japan.

Shiba Yoshimasa (1350 -1410) describes as part of what was later called Bushido, the Way of the Warrior, that one

should think and then act not only his own fame, but also that of his descendants.

This is something, that would have limited the choices of many a samurai, as they were expected not only to do what they can do best, but what is best for their family. So the modern idea of individual fulfillment was not as prevalent back then (although to some degree it did exist). Luckily for those, who were inept fighters, there were other ways to build a good reputation for yourself and your family. Or disgrace, for that matter. Shiba Yoshimasa suggests to practice all of the arts. He writes:

how deplorable it must be to be among people reciting linked verse and to have to make some excuse for one's inability, or to sit chin in hands while others are playing music together, or to be even unable to join in the beginning of a game of football.

Yoshimasa lists the recitation of chants, poetry and musical instruments as arts, every samurai should pursue. Also writing, especially when courting a young lady, is best done by oneself instead of a scribe. He further lists the pasttime games such as go and shogi, as well as archery. Then he tells us, how his parents educated him in all of these, and how it had helped him avoid embarrassment, despite not excelling in any of them in particular. So if you arrive at a decent level of fighting skills, so as to not embarrass yourself in front of others, that would have been sufficient.

Sadly, he then goes on to complain about "today's youth", which was probably his intention for writing in the first place. The fact, however, is, that this was the way, noble children were educated during his time and also later.

Of course, in times of war, fighting was more essential than playing music. This applies less to later times (Sengoku period included, to a certain degree), when military tactics changed and many peasants were drawn into the army, where being an actual warrior became less important than knowledge of military tactics like Sun Tzu's Art of War. Of course, even before that, fighting was only one way to solidify your position, diplomacy for example being another.

The "justification" to have a warrior class was an issue during the late Sengoku period, the so called Azuchi-Momoyama period, when things started to quiet down. This is seen as the reason, why the Shogun all but in name Toyotomi Hideyoshi started two failed invasions of Korea, so as to keep the warrior class busy fighting. During the Edo period, without any wars, the warrior class seemed even less necessary, and so they did even more art, martial as well as non-martial. Their role in society was mostly that of administrators.

To summarize: Being a Samurai was much more than fighting, and you were expected to know more stuff anyway. Also, with times, the role of the warrior class changed a lot and required different skill sets.

Source: Shiba Yoshimasa's writing on this subject was translated by William Scott Wilson in his collection Ideals of the Samurai: Writings of Japanese Warriors.

EDIT: In regard to Bushido.

Although there were groups of people who did lots of fighting throughout Japanese history, "samurai" is a catch-all word that applies to people who lived radically different lifestyles depending on the time period. When someone says "samurai," they are probably thinking of a blend of the fighting man of the Sengoku Period and the upper class of the Edo Period.

Moreover, the class system that people think of regarding pre-modern Japan, is actually pretty much only relevant once the Edo period starts in the early 17th century when Neo-Confucianism exploded in popularity during the time when a new social order was being sought out after centuries of war.

Although Confucian thought was imported to Japan much earlier along with other Chinese ideas and products, it basically wasn't considered very useful as a philosophy until society needed a way to reorganize itself in a way that had continuity with but also broke from its past.

Neo-Confucianism was it, and on particular branch was eventually made a part of the state by the Kansei Reforms of 1790.

With Neo-Confucianism came this idea that there would be 4 castes of people and social mobility would be limited. It's a relatively late idea in Japanese history, and most notably only existed in a time of incredible peace. So if a warrior did not want to fight during this period of limited mobility, that's just fine. There was no fighting to be had, but plenty of other tasks to attend to, from local administration to government office to involving oneself in the arts to philosophy or poetry.

Before that time, things were less ambiguously defined, and there were periods of great social mobility. In fact, one of the three great unifiers of Japan at the end of the 16th century, Toyotomi Hideyoshi, was of low birth, and he went on not only to become a distinguished warrior and general, but also the most powerful man in the country. It should be noted, though, that this was rare, but it was by no means impossible for a man of small means to find access to weapons and then work his way up in society through skill of arms during the endless fighting of the Sengoku Period.

But, to answer the spirit of your question, during these centuries of war, what would happen when the child of a warlord was unwilling or unable to fight? Well, for one, keep in mind that what you are considering here is actually a very small class of elites. Most fighting men were conscripts and not professional warriors. So, while the child of a Daimyo would be expected to be capable of fighting, there was a place for high-ranked warriors whose talents better applied to strategy or tactics or the convoluted politics of the era than actual hand-to-hand combat. A great example would be Yamamoto Kansuke, one of the advisers to one of the greatest military leaders of the Sengoku Era, Takeda Shingen. This warrior was said to be lame in one leg and have lost one of his eyes, but was most valued as a tactical commander for Shingen's armies. In fact, he pulled off a brilliant pincer maneuver (the same one used by Hannibal at the Battle of Cannae) in the 4th Battle of Kawanakajima in 1561--although he didn't live to see the success, as he suicide charged into the enemy ranks when he mistakenly believed it to be a failure.

Even so, if a military commander did not have the respect of his or her men because of talk of cowardice, they were not likely to remain in a high position for long. Keep in mind also that children as young as 13 might fight in battle; families would know quickly whether their children were completely unsuited for their position. In that case, becoming a monk was common, as was sending that child to be married or adopted into a lower ranking family and thus making them somebody else's problem.

To bring it back earlier, when warfare was not a constant but a warrior class of sorts existed, children who were not suited to be warriors tended to find themselves sidelined when it came to succession. Although primogeniture was the common practice, it was not a rule, and all sorts of shenanigans involving adoption and marriage and disinheritance and monks were used to arrange the line of succession however the head of the family (or a rival) wished it. Having a son unfit as a warrior may have been a convenient reason to push him out of the line of succession; of course, other criteria existed for other families.

But such children eventually gradually lost their status over the generations, more often than not devolving into a nascent middle class devoted to the management of landholdings under the public land management and taxation system of Ritsuryou, or it's private Shoen equivalent, before it was all exploded by later social and political upheaval.

And to bring it back to the origin of the samurai, that comes from the formation of five warrior clans during the Heian Period. These clans were subfamilies of the Imperial House, and to keep politics and succession from getting too complicated, remove them from the Imperial House's expenses, and find something for these families to do, they were officially designated as the first warriors by birth and trade in Japanese history. They were then marched off into the wilderness to fight barbarians, and so much for them--at least, you would think so. One family in particular came to dominate the politics of that time, but that's another story for another time.

At any rate, these families had close ties to the nobility of the Imperial Court, and if fighting was not possible, there were other options available, usually involving entanglement in the nobility and politics.

As a side question, it seems to me that most "warrior castes" are also ruling castes as well. The only counterexample I can think of are the Janissaries, and they became part of the ruling class too (as I understand it, please correct me if I'm mistaken).

So, my question is: have there been been any historical examples of warrior classes who weren't also the ruling class?

You probably weren't thinking of republican Rome, but the upper classes were very much a military ruling class in the vein of their Hindu or Japanese brethren, and I think some examples from the period ate illustrative.

Cicero and Augustus were well known to be at best ambivalent and incompetent military leaders, and at worst were often accused of cowardice. But military service was part of rising in Roman politics. Cicero got his out of the way as quickly as possible, and went on to a brilliant political and legal career. Augustus carefully stage managed his battles to get maximum credit for what others (Agrippa) did while mostly avoiding actually leading men in battle. He was later literally worshipped as a God.

These men managed to rise in a military oriented ruling class while being bad at and uninterested in fighting. They essentially treated military service the way college students in stem treat gen ed requirements, they either get through them quickly so they can focus on their real interests or they get someone else to do the work.

Also, keep in mind that in peacetime military service in a warrior caste really just meant keeping your armor clean and the odd parade.