How can I prevent my PC from tripping the ARC fault breaker when turning it on?

AFCI senses RF noise in a circuit, such as generated by an arc to ground, and turns off power to prevent a fire. They can be false triggered by electric motors which have arcing at the brushes. AFCI ALSO does GFCI at somewhat higher current, ~35mA IIRC vs. the 5mA for a GFCI. Still probably enough to prevent a lethal shock which typically takes 100-300 mA.

Switching power supplies generate RF noise, maybe enough to trigger this AFCI. The AFCI could be defective, you could have a leak to ground in your PSU, or the PSU could be generating a lot of RF noise.

Try plugging in your computer with a 'cheater plug' that temporarily disconnects the ground, so the GFCI feature of the AFCI can't pop the circuit. You may need to cover the ground screw in the center of the outlet to be sure you have defeated the ground. If this stops the problem, the PSU is defective with a potentially fatal current leak to ground!!

If that doesn't work, try putting a surge protector (REQUIRES a connection to ground) that clips voltage spikes between the computer and the plug. This may filter out enough RF to fix the problem.

As others noted, plugging into a UPS, which usually has noise cleaning circuitry may clean up the RF from the computer enough to stop the problem.

There are also dedicated EMI/RFI filters that 'clean up' 'dirty' power. I had one computer that kept rebooting till I put an EMI/RFI filter between it and the world.

Try the computer on a different AFCI circuit. It's possible the AFCI is just flaky. If you have another circuit in your apartment where the computer works, and the landlord won't replace the AFCI breaker (they're about $50) try a long 14 or 16 GA extension, usually sold for exterior use. Less than ideal, but you just want something that lets you use your computer.

Let us know what happened?

There are two possibilities that come to mind here:

1. You're actually over current somehow.
2. There's an issue with your computer's PSU that's causing it to trip the breaker.

The first case is relatively easy to check out, get a 1000W electric space heater, plug it in in place of your PC, and see if turning it on trips the breaker. If it does, you're over current for the breaker, and thus either need to put the PC on a different circuit, or figure out what else is contributing to the power draw enough that you're over current. If you're on 120V mains power, then this is actually possible as perfect efficiency at max draw would translate to about 11A (factoring in the efficiency of the PSU given the 80 PLUS gold rating), so you would only need 4A of additional draw on the circuit to it the trip point.

If that is not the case, then the issue is probably your PSU causing a power fluctuation on startup that looks to the breaker like a fault. Getting a better PSU should help here, but you can probably also get away with using a UPS for the PC (which should smooth over such spikes because of how they work internally). Actually testing this properly is complicated and expensive though.

On a slightly related note, a 1kW PSU is almost certainly overkill for your build. Even a conservative estimate of peak power requirements for the stated components comes out to about 450-500W (so at most a 600W PSU to get right about the ideal spot for efficiency most of the time). I've seen a number of cases over the years where excessively over-specced power supplies would have issues with introducing noise or odd glitches/spikes on the mains circuit they were connected to, so this may actually be a contributing factor (and you could probably reduce waste heat from the PSU by using one rated closer to your actual power requirements)