Why 22,500?

Anyways based on William Farris' population estimates, a population of 22,500 would be one of Japan's small-to-medium size province. Which means the answer depend on 1) just who you were and 2) exactly when and where in the Kamakura period we're talking about.

In the Kamakura, the nominal head of a province was the kokushi (provincial administrator) appointed by the emperor's court in Kyōto. They would have been in charge of running the public land of the province and the many shōkan (estate administrators) appointed by the court to run Japan's many shōen (estates).

However, shōen in eastern Honshū were ran by jitō (local heads), who were appointed by the Kamakura Bakufu from among the ranks of Kamakura's direct vassals in the province, called gokenin (men of the house). Among the head jitō of a province, a shugo (protector) was appointed by Kamakura to oversee the security of the province. The goseibai shikimoku, Kamakura's first set of laws issued in 1232, formalized the shugo's duties as arranging the province's gokenin to report for duty at Kamakura or Kyōto and dealing with rebels, murderers, bandits, and pirates.

For the actual forces, shugo mobilized from the ranks of gokenin. The gokenin themselves would bolster their ranks by bringing along other provincial warriors and strongmen who could be mounted but were more often on foot, brought along by personal ties often but not necessarily familial. The number of these people the gokenin brought along seem to have varied greatly, from just a single person to close to 20 men. A 1272 roster for Izumi province however gave 19 gokenin and 79 followers, for an average of just over 4 followers per gokenin at 98 men total. For an aborted invasion of Korea following the first Mongol invasion, it's also recorded that the gokenin mobilized 5 followers. For a separate era, according to the Kōyō Gunkan, the Takeda samurai was estimated to be able to mobilize 4 followers on average. So 4 or 5 per gokenin should probably be the norm for active campaign duty.

In theory, the kokushi should have been able to mirror a shugo's mobilization structure. In practice, the only time it was (probably) done was in the Jōkyū War in 1221 when Kyōto fought against Kamakura. After Kyōto lost the war, a lot gokenin were transfered to western Honshū and Kyūshū, and a lot of shōen were given to them for control. However, a lot of shōen remained in the hands of the shōkan. Just what the ratio of jitō to shōkan is a mystery, complicated by the fact that many jitō were also recognized as shōkan. Likewise the ratio of gokenin to non-gokenin local strongmen is also a mystery, for which we only have incomplete fragments and examples not at all generalizable to the whole of Japan.

Even after the Jōkyū War however, the shugo could still only mobilize gokenin. It wasn't until after the first Mongol invasion that the shugo seem to have gained the right to mobilize non-gokenin warriors, at least in Kyūshū in defense against the Mongols.

So what forces you had at your disposal depended on who you were. If you were a kokushi, in theory you had the entire province's forces at your disposal, but in practice you had none. If you were a shugo on the other hand, it would depend on exactly how many gokenin were in your province, what ratio of gokenin to non-gokenin made up your province's local strongmen (non-gokenin likely could not mobilize as many men as gokenin), and whether or not you lived in a time and place that allowed you to mobilize non-gokenin.

Just to give one example, Faris gives the population estimate of Awaji province of around 20,500, based on its 1223 land register and taking the assumption the register only recorded rice farms instead of all farms (only 13 such registers survive in enough detail to do such estimates). As previously mentioned, neighbouring Izumi province mustered 98 men in 1272. Both were labeled as "small provinces" in the middle Heian, and since they are only separated by Ōsaka bay, one can hope that they're the most comparable. If both provinces had the same number of population, gokenin, and support warriors (and that is a huge if), then that would be about 0.44% of the population. That's actually nothing to scoff at considering these men were sent half way across Japan to guard against the Mongol Invasions. Just for comparison purposes, using wiki's figures and by no means meant to be accurate, the USA 1860 census put the population at 31,443,321. Roughly 180,000 men were involved for the battle of Gettysburg, the largest battle of the American Civil War. When compared to the 1860 census that's 0.57%.

On a final note, the question presuppose there was a provincial ruler, probably on the incorrect assumption that Kamakura Japan was "feudal" or that it was divided into domains like in the Edo period. This was not the case. The shugo was not the province's "ruler" per se. They did not have administrative or legal authority. However, as the kokushi's de facto control was severely limited by not being able to call on the gokenin, neither was the kokushi in reality. Reality was a confusing (to us) web of co-operating and competing authorities.