Having a HUGE capacitor bank is not a wise idea as you will be challenged to afford the capacitors and it wont buy you much brown out time. Get a UPS or a power supply rate for MUCH higher power than you need... but judging by your $ estimate of your PC it probably needs at least 500W and a 1kW supply does not buy you much extra time for a power dropout... "maybe" a half cycle.
get a live UPS, not standby, so it always runs from battery while the AC charges the battery thus no battery current is required. Plastic RF cap on battery may help if transient EMI problems.
do you know the advantages of a UPS?
...not to mention design for lightning surge protection, power cycle current limit, thermal protection, ESD , line noise filter and other transient protection....
If UPS is ok but still get power cycle on PC-PSU on transient outage, then consider if it is radiated or conducted noise and isolate the problem then solve it,
- transient radiated interference is common on poor system designs due to large surge currents on power-up.