Congrats Fire on you trip to Japan!
I still haven't been in Japan, so I don't know much, but I still think these few things I know might be helpful
If you're going to Japan early April, I think there may still be sakura blooming in the northern Japan. By typing 'sakura forecast 2024' you should be able to see a forecast with a map that shows when sakura starts blooming (looking at blooming sakura is called 'hanami' 花見, Japanese love to do it every year, hence the blooming forecast). The photos each year look absolutely stunning, so it would be sad to miss, especially since you're in the right time for that. I've also heard that sakura blooms only for few days, so if you decide to see it, the timing is of the essence (though I'm not sure how true is that, I haven't been there).
Another seasonal occurrence is the rainy season, 'tsuyu' 梅雨. I've read it starts early June, there may be some variations every year, I'm sure there is a forecast for it too. When it starts, it will heavily rain all day, all night, all day and so on. Your trip might end before the rainy season starts, then there will be no problem. If it starts earlier while you're still in Japan, I think it will be best to plan some indoor attractions for these last few days
About food, my japanese teachers said that it is a common occurrence to have an upset stomach for first few days after coming to Japan due to difference in bacterial flora - it might be a good idea to take some medicine for that.
I also think Osaka is famous for its takoyaki たこ焼き, but it might be a common knowledge
I'm not able to answer if shopping for men's clothes is ok culturally. I think their first thought might be that you do not know that these clothes are for men, and the second that you might be an otaku? I don't know how ok it will be, though I doubt they will say anything to you anyway. But I think there may also be another problem. Asians have a very, very slim build. Depending on your genetics, on your build and on clothes style, these clothes may be too slim. Especially men's clothes - I have 180 cm and as a woman I am not able to buy any men's clothes in Poland where I live, because my hips are too thick.
My japanese teachers didn't say anything about buying clothes, but they warned me about buying shoes - the biggest shoe size for women in Japan is 'L' and it is equal in shoe length to european size 39, but these shoes are so narrow they don't fit at all. And buying it anyway means wounds on your feet - yes, they did buy it...
Similarly, hair dressers or japanese hair dyes might be a bad idea, though that also depends on your genetics - Japanese have a very thick and straight hair, so their dyes are very strong and destructive for less thick hair, they are also cut and did very differently. My japanese teachers always said to ask a hairdresser in Japan if they did non-Japanese hair beforehand because of that.
When you're in any bigger city, like Tokyo, I know there are hidden gems there. Very well hidden, in a very narrow streets, so narrow one person barely fits there

Look for Hanabusa Inari Shrine in Tokyo on youtube for example - I've heard there are many shinto crams like that, in various cities. Because these cities expanded while there were already some shinto shrines, so they were 'absorbed' by the very tall buildings around them. I think it might be magical to visit at least this one shrine

And it might be worth it to sometimes go into narrow streets like that.