Can the AK400 cool the 5800x3d quietly and good?
No.
The 5800X and 5800X3D have very high power
density - the 105WTDP (which is 142W under full load) all comes from a single tiny chiplet, unlike the 5900X and 5950X which spread that same 142W over two chiplets.
If you don't mind high temperatures (85C all-core) and marginally reduced performance (we're talking boost speeds of 50-75MHz lower than if it was running below 70C) then a basic 4-heatpipe tower is fine. You're never going to get it running cool enough for it to not dial back boost clocks a bit, but 85C temperatures are acceptable and you can reasonably achieve those even with basic coolers. I've had NH-D15 and AK620 on a 5800X and even those struggle to keep temperatures under 80C at full load.
I'm using a dual-radiator Alphacool loop, I struggle to keep temperatures of a 5800X under 70C because it's asking
a lot to remove the lion's share of 142W from a tiny concentrated part of an already tiny die. About 70-80% of the peak power draw in Zen3 comes from the core logic, highlighted in yellow:
and here is where the 70-80% of the that 142W is coming from, in more detail:
The problem isn't too much power for coolers to handle, the problem is too much heat in a tiny space to keep temperatures low.
IMO, get a slightly larger air cooler - the ID-Cooling SE-226-XT or the Thermalright PA120 - simply because 142W will need dissipating and a larger cooler with more heatpipes and more surface area is the only way to do that
quietly. As for temperatures, you're going to need a good 280mm or 360mm AIO at a bare minumum to bring those down by any meaningful amount. If you're on a budget my suggestion to you is to just ignore the temperatures - the 5800X3D will easily reach its own thermal limits with normal boost behaviour and dial back clocks slightly under all-core workloads. That's not a problem, it just means you're only going to get 98% of the performance you'd get with top-tier cooling.
It's absolutely not going to overheat, even with a 4-heatpipe slim tower cooler, but for an extra $15, one of the larger coolers will get you more of the potential performance and give you thermal headroom to reduced the fan speed. Even with one of the suggested large tower coolers, you're still going to have the fan running at 100% on the default fan curve because it'll rapidly exceed 80C in an all-core load and almost every motherboard fan curve defaults to full speed by the time temperatures are up at 80C or so. IMO you should set the fan curve to 100% duty only at 100C, and have the "expected" full-load temperature of 85C running at 60-75% fan speed. That way it still has headroom to speed up if it gets silly-hot, but it should be much quieter when running an all-core load: