I'm not well-versed in Japan's PC markets (although I've seen pictures and videos of Akihabara), but I'll give it a shot.
I would go with an AMD 2600X, and an X470 motherboard. (There may be an X570 motherboard coming out soon, and the Ryzen 3000-series / Zen 2 architecture CPUs should be out, sometime later this year. Maybe a few months from now though, if you need a PC NOW, then build now.)
Get a 2x8GB (16GB kit) of DDR4-3200 memory, GSKill or Corsair, maybe Team or Adata if you want to save money, but those kits often don't run at their advertised speeds in AMD AM4 rigs.
And what resolution does your Son want to play games in? If 1080P, then look at getting an NVidia RTX 2060-based GPU graphics card, they just debuted yesterday at Newegg, they're around $350-400 USD, not sure how many YEN that would be.
And get a DECENT keyboard and mouse combo. (Very subjective, based on hand size, reflexes, and low-sens / high-sens and claw-grip versus other mouse gaming styles.)
I can't comment on screens, but it's rumored that NVidia has now started to add support for AMD's FreeSync-compatible VRR monitors, vis-a-via VESA Adaptive Sync feature, and NVidia is branding it as "GSync Basic" or something like that. Maybe wait a month or two for monitor maker's branding to update their boxes (if nothing else, with stickers), that indicate GSync Basic support, so you know that the monitor will be supported by your NVidia card?
I don't game (much) anymore, used to be pretty hardcore 20 years ago. I have 4K UHD TVs as screens, but don't game at 4K. My understanding is that most "serious" gamers want a high-refresh-rate, variable-refresh-rate monitor. Which means, a 144Hz 1080P / 1440P screen, that supports AMD's FreeSync or FreeSync 2, preferably over HDMI, although DisplayPort is also acceptable in most cases.
Edit: And you mentioned budget as being important. If performance is more important that budget, and you have to build more-or-less "today" (right now), then an Intel 9600K or 9700K rig might be an option instead, but be aware that: 1) It will be more expensive, sometimes significantly so, and 2) AMD's AM4 platform will be getting "Zen 2-based" CPUs later this year, which are rumored to eliminate Intel's current single-threaded performance advantages, and is upgradable, while Intel's series-300 socket 1151 platforms are basically a total dead end as far as upgrades go. That's why I suggested an AMD build, and I run AMD AM4 rigs myself.
If you do choose to go Intel, get a Z390 mobo, rather than a potentially-cheaper Z370 board.