TL;DR. Establish a strong early foundation with whips, build lots of siege, start conquering as soon as UHV1 is done, develop Kyoto/Edo for stability, conquer necessary territories when weak, keep Foreign cities low in pop, Specialist Economy.
I'll preface this by saying that I have not played Japan on a 600AD start in a while, because I find the first goal pretty tedious and formulaic after you 'solve' it, and I greatly enjoy Japan's relative starting position in the 1700AD start better: I find it gives a better gameplay experience and challenge than the 600AD scenario, where a solid early start trivializes everything.
That being said, in the 600AD start, the first thing that affects how you do UHV2 is whether or not you chose to conquer Seoul early and therefore whether it was included in your UHV1 requirements. Assuming you settle Kagoshima/Nagasaki for the luxuries, a Seoul conquest means you have 4 cities instead of 3 to average your culture across (Edo (1N of river to work silver + Kyoto for core). Seoul alone can generate enough Great People and commerce to make up that difference; the more important part is that you need to spend the first 20 or so turns on whipping galleys and catapults to conquer Seoul, rather than building up infrastructure. If you choose to conquer Seoul, it needs to be done ASAP. The benefit of doing so is that you get another solid city, along with Kyoto/Edo, to fuel military and commerce. Kagoshima/Nagasaki are weak cities, and are basically there just to secure the luxuries.
Regardless of whether you go for Seoul or not, civics should be swapped to Despotism/Citizenship on first turn; the other columns don't really matter. Despotism is needed because Japan (and Korea) have basically no hammers, a lot of food, and a lot of luxuries. Whips are therefore the best way to get infrastructure and military. Citizenship is used because it's fudging busted, and synergizes perfectly with UHV1: several cities which need to grow and develop a lot of infrastructure. There is also only like 3 plots in Japan/Korea that can be farmed, which means Vassalage is worthless. Make sure to focus culture bombs on the GP farms; higher culture levels improve specialist yields. I like to use them as soon as possible, as waiting for the Renaissance Era so that they have better yield per bomb really just takes too long, and is offset by having to wait that long before starting conquests. With Medieval Era culture bombs, you usually need about 4 - 8 Great Artists (off memory), which is easy to accomplish if you're running as many specialists off your big cities as possible. Himeji Castle works out to about one Great Artist.
The goal is to clear UHV1 as soon as possible. Since it's a 'by' goal, you can have it done well before the actual deadline, and once it's complete, start conquering stuff right away. Switching to Conquest (for the stability bonuses) and anything but Redistribution following UHV1 will help with stability. As with any conquest goal, the name of the game is lots of siege units. With the addition of 6 or so trebuchets and 4 galleys, your starting army is enough to conduct basically all your conquests. Korea is the easiest: if Seoul is already conquered as part of UHV1, then you might have Korea vassalized, or collapsed. If not, they're usually in shambles by the 1350's, and not a challenge. Manchuria tends not to be settled (minus Shenyang), so just throw several settlers into the area. Manchuria is also the only area in your early to mid expansion that actually has hammers, so it can be used to produce new military if needed. Kyoto/Edo, after producing the additional siege/navy, focus on commerce.
China is the main variable with a 600AD start. They either fall apart to the Mongols, or beat them and become a juggernaut for a few centuries. In my playthroughs, I preferred the second option. Even after beating the Mongols, I never saw the Chinese able to stay alive past 1600 or so, and would collapse to indies, leaving easy conquests. If they collapsed to the Mongols, however, they get a scripted respawn as the Ming in around 1400, and become a stable and viable competitor and obstacle. In that case, pumping out more siege and military is enough anyways. Use whips and such where necessary. Conquer only what is necessary in China to achieve the goal: Beijing, Nanjing, Hangzhou, and Guangzhou tends to be enough. If China hasn't collapsed after that, conquer Xian (likely their capital) and then release it to indies or donate it to Tibet or something. For stability purposes, the Chinese cities will be whipped and kept under their food maximums.
The rest is easy. I find that the Europeans in 600AD start do a very poor job at colonizing, really only working due to the scripted spawns, and generally spawn stacks with Arquebruisers at best. Even with only Manchuria, Korea, and Japan, there's absolutely no difficulty in conquering the required South Asian territories. Start with Thailand: Thailand is weak militarily, collapses when Ayutthaya is conquered, and Ayutthaya itself tends to be a great city to fuel future conquests and techs (high commerce and production). Philippines can be conquered once Spain spawns Manilla. Indonesia can be mopped up when the Dutch spawn their conquerors. Otherwise, just focus on maintaining stability. After conquering Manchuria, avoid whipping out of Kyoto/Edo for military as much: they need to maintain their populations as high as possible, as the only two core cities in Japan until the Industrial Era. Whip as much as possible in China and Southeast Asia to minimize populations. Korea, Japan, and two or three select supercities in China/Southeast Asia will provide enough commerce to win UHV3.
Honestly, in the 600AD start, you can conduct the necessary conquests long before Zaibatsus come into play, which is why I find the 1700AD start much more fun. Your starting units, supplemented with siege and some more stuff, are more than enough to conquer everything. It really just comes down to a waiting game after China + Manchuria + Japan + Korea, swooping up the Southeast cities when they're held by weak defenders. With the four East Asian territories, you should have enough beakers that UHV3 is a joke. After UHV1, prioritize Kyoto/Edo/Seoul to run Scientists and Merchants, make Academies and settle Great Scientists in them, and you should run obscene amount of beakers to offset Japan's bad tech modifiers. Use Egalitarianism/Free Enterprise when they are available.
Again, this is all based off some of the older Git revisions. I have not tried a 600AD start yet with the changes to conquest stability, nor with the new AI collapse to core mechanics. However, the fundamentals should still hold true: UHV1 done ASAP, then conquer.
Edit: Moved TL;DR to top.
