It's gonna be... interesting.
First question will be if the Emperor can survive long enough to continue being relevant. It would not be surprising for literally all of his rivals to try to make a shot at him since he suddenly lost a lot of his support. Remember, St. Nicholas of Japan was once literally almost beheaded by a Samurai not long after he started his missionary work, and he was just a minor missionary at the time. Let's say St. Nicholas of Japan is significantly more successful in evangelizing the Kirishitan population into the Orthodox Church, while a contingent of Orthodox Samurai lead by Takuma (Paul) Sawabe are able to keep the Emperor alive long enough for some of the local governors and population to actually convert. I'd rank this as Barbarossa level implausible rather than Sealion level impossible. This opens up a host of new problems.
Problem number one is going to be what Patriarchy Japan will be a part of. As most people who study Orthodoxy know, the Orthodox Church is divided into a series of patriarchates that all coexist within the Orthodox Church as a whole, with the Patriarchate of Constantinople being first among equals but otherwise holding no extraordinary power compared to the other patriarchs. The minute Japan converts, there will be a few factions that will show up:
1.) The Russian Patriarchy. The Empire of Russia is going to be extremely... interested by this new Orthodox power to their eastern border. While initial relations will undoubtedly be unwavering cordial, eventually the Patriarch of Russia (which, at this time period, unfortunately became very keen on Russian Imperial interests) may claim that since a Russian Orthodox priest converted Japan, that the Spiritual Authority of Japan lay with Moscow, and perhaps even the temporal authority. A quick glance at Ukraine shows that this can lead to some...unwelcome consequences. While if Japan goes this route it will become very very close to Russia and Russian interests, Japan will now have the massive bulwark of Russia to assist them, assuming they are not subsumed.
2.) The Patriarchy of Constantinople. The Patriarch of Constantinople at this time is living as an exile in his own nation, with the Ottoman Empire having plowed over most of the Balkans during this time. Even worse, the Greek War of Independence lead to deteriorated conditions for the Patriarch. He is confined to a Gothic style church built in Istanbul, deprived of most of the beautiful ancient Byzantine churches that former patriarchs had access to. Even worse, the Ottoman authorities had a bit of a...penchant for "somehow winding up" with a lot of tithes from the faithful. With Japan as a part of the Patriarchy of Constantinople, the Patriarch will now be able to actually gain important things like money and even some power to be able to perhaps win the Greeks of the Ottoman Empire some more rights. However, paying tithes to an exile half a world away to help a group of people who the Japanese might not have ever even had any contact with might loose Orthodoxy in Japan.
3.) An Autocephalous Patriarchy of Japan. This one is probably what the OP is thinking will happen. Perhaps St. Nicholas of Japan joins this new church as a leading bishop (though I would bet money that St. Nicholas would never allow himself to be the patriarch) and helps the church get established. There will unfortunately inevitably be far more syncretism with this option than otherwise. However, it will be able to spread faster with the average Japanese.
Honestly, no matter what happens, Europe will not treat Japan any differently. To put it into more blunt words, the average European does not give a hoot about the average Orthodox peasant no matter how Christian they are. The Protestant churches viewed Orthodoxy as no better than Catholicism, and the Catholic church views the Orthodox church as backwards schismatics who need to be "gently" brought back into the fold, at least that is the viewpoint of the more politically devout of the two denominations. No matter what, the fact is that race matters more than religion in the 19th and 20th centuries. The Ethiopian nation has been Christian longer than many parts of Europe; this didn't stop Italy. Neither would Japans orthodoxy stop a prospective European power.
Really, my prediction is that Japan is not able to expand past Korea under Orthodoxy. While what territories Japan would have held would become far more integrated as Japan looks inward to enforce Orthodoxy, there would be no Russo-Japanese paneurasian empire.