Do you actually need a case? You mention having a PSU and DDR3, and I'm just wondering if you happen to have a suitable case floating around as well. Even if it doesn't look particularly cool, that would be another $35+ that you could put toward other components, which in a system at this budget level could make a big difference.
Also, do you need a wireless card, or might it be practical to run an ethernet cable to your computer? That would of course depend on whether you have a router to connect it to nearby, or if its in some other room.
Without a case or wireless card, something like this might be an option for around $400...
PCPartPicker part list /
Price breakdown by merchant
CPU: AMD - Ryzen 3 2200G 3.5GHz Quad-Core Processor ($96.99 @ SuperBiiz)
Motherboard: ASRock - B450 Pro4 ATX AM4 Motherboard ($59.99 @ Newegg)
Memory: Patriot - Viper 4 8GB (2 x 4GB) DDR4-3000 Memory ($70.98 @ Newegg)
Storage: Team - L5 LITE 240GB 2.5" Solid State Drive ($37.99 @ Newegg)
Video Card: Gigabyte - Radeon RX 570 4GB Gaming 4G Video Card ($139.99 @ Newegg)
Total: $405.94
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2018-11-04 01:50 EDT-0400
If you intend on gaming, an RX 570 would be a lot faster than the 2400G's integrated graphics, getting you around 3 to 4 times the performance in most newer games, provided the CPU can keep up. So I would look for ways to try and fit one in there. I agree about avoiding the older FX processors though, as they can limit performance in many games. If you absolutely can't work an RX 570 in, it might also be worth considering an RX 560, but it will be about half as fast, with performance roughly halfway between an RX 570 and the 2400G's integrated graphics.
I went with a 240GB SSD with no additional hard drive in that example, but it would be tight and you would only be able to keep a handful of recent games installed at a given time. Doubling the SSD's capacity for another $20 as in the previous post would probably be better than getting a separate HDD and SSD though. Or, if you absolutely had to, you could stick with a hard drive alone, as it will generally just affect load times as far as gaming performance goes. An SSD would also make your system feel more responsive outside of games though.
I also went with DDR4-3000 RAM for about $10 more, as it will help Ryzen's performance (especially with integrated graphics), though if you are getting a dedicated graphics card, then the performance impact won't be critically bad by going with somewhat slower memory. If cutting back to DDR4-2400 to save $10 helps you to get a better graphics card, then it might potentially be worth the reduction in CPU performance.
If you have absolutely no intention to overclock, you might also consider cutting back to an A320 motherboard, for as much as $20 less. Generally, I wouldn't recommend that for a system used for gaming, as a Ryzen 2200G has a decent amount of room for overclocking, and being able to overclock may keep your system relevant longer and help it better run some of the more demanding games, but with a budget like this, the option for a non-overclocking budget motherboard might be worth considering.
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