The Russo-Japanese war can be described as the first modern war lost by a modern civilization. How did the West cope with the fact that a presumed inferiour race defeated a 'white' imperialistic power? Are there any notions of racial slur against the Japanese? Did the west got scared of the 'Yellow Peril? And how did the press describe this Asian culture?
There were mixed responses to an Asian power defeating a Western one, but both positive and negative impressions of Japan took into account its rising status. The concept of the "Yellow Peril" shifted over from the Chinese to the Japanese in America, but it would be wrong to discount the many voices in the Anglosphere that cheered Japan on. Also, the Japanese were already held with respect to some extent in the United States at the time.
https://muse.jhu.edu/article/241933
Politically, opinions were divided based on the already-existing balance of power in Europe, namely if your country as pro or anti Russia.
The United States and Japan had a "gentlemen's agreement" in 1907, which refrained from openly restricting against Japanese immigration (instead of outright banning them), as a nod to Japan's demands for international equality. Of course, this happened at the same time as nativist sentiments were rising.
The British reconsidered their strategies in the Pacific, and strengthened their holdings there, but they also had positive feelings towards Japan, already having signed the Anglo-Japanese Alliance treaty in 1902 to contain Russia. Much of the news and fiction written in England during the war was on the Japanese side. On the other hand, British intellectuals were troubled that this might inspire Indian calls for self-rule.
Russia was humiliated and destabilized by their defeat, which felt worse because of the race of their enemy. The Tsar and his government suffered a drastic hit to their prestige, and the revolution in 1905 was partly affected by the war.
The Germans were encouraged by this sign of weakness from Russia, and became much more belligerent towards France and other Russian allies.
Overall, it led to a slight weakening of European prestige, and people began to consider global politics as incorporating more of the world than just Europe. Socialists, anti-colonialists, future Asian nationalists like Gandhi, Sun Yat-sen and Nehru, and a few subject European peoples like the Irish and the Poles felt encouraged by what they saw as a crack in the current political system and notions of racial supremacy.
The House of Morgan (by Ron Chernov) documents what US bankers thought of Japan in the 1920s.
Basically the Morgan bankers were very optimistic about Japan. They saw it as an economic and cultural success story, calling it "The England of Asia". These bankers idolized English culture, so it was a real compliment. The Morgan banks had strong faith in the liberal elements of Japanese society, and maintained the Japanese state as an important client long after the militarists got control and started murdering the liberals the bank used to work with.