Joe,
ARGH. Copy/paste destroyed my Word formatting and it really bothers me. I hope you can still follow this post.
I have all UT settings maxed out, screen resolution is set to the monitor's native 1680x1050. UT stays capped at around 80+ fps with multiplayer. Single player looks to average about 105+. There is a line in the ini file that caps multiplayer refresh rate to 80 fps but it doesn't matter since my monitor is only 60 fps. We can certainly take some multiplayer comparison shots the next time I am online, which is usually Saturday and Sunday afternoon around 4PM CST.
You mentioned that you are running XP. I will make some assumptions and point out some problems.
1. Running XP? You probably won't be able to find a compatible driver for the RX580 and may not be able to take advantage of its full potential. Without a proper driver you may be stuck with low resolution and poor fps.
2. Is your processor old? It will bottleneck way before the video card will. If your processor does not max out during game play then you are ok and can enjoy a higher resolution, higher FPS and higher settings up until the processor hits 100 percent. Then you will want to lower resolution and/or settings until you balance processor and GPU utilization. Otherwise, you may experience stuttering. My old rig had an AMD Phenom, which had no issue running UT. The video card, Radeon HD 4850, was actually the bottleneck.
3. Low system memory? That's ok. Assuming XP is 32 bit, you are limited to about 3.5GB’ish of memory, which is enough. My old rig was Vista 32 bit with 3GB ram and it never maxed out ram.
4. Does your motherboard have a PCI express x16 slot? You will need one. Don't worry if its v2.0 or v3.0. That just indicates data throughput and should be reverse compatible.
5. Old, underpowered PSU? A couple of problems:
a. PSU is old, therefore the electrolytic capacitors in the PSU are old and may be drying out. Higher GPU load will likely introduce more voltage ripple which can cause issues.
b. Is PSU under powered? I believe the manufacturer recommends 500 watts to run the RX580. I believe the manufacturer covers their butt by over estimating. You can likely get by with slightly less but I wouldn't want to try. The PSU power efficiency curve likes to be between about 40-80 percent of max power. Going lower is ok. You just lose a bit of efficiency. But I personally would not want to push a PSU past 80 percent. There are PSU calculators online that might help you. See if power requirements of your current video card is comparable to the RX580. If you bought the PC built, manufacturers usually install junk PSU's that barely meet minimum power requirement. Check the PSU label.
c. Old motherboard? The electrolytic capacitors on the motherboard are likely drying out and will exacerbate any issues caused by an old worn out PSU when you introduce a heavier load. Do you see electrolytic capacitors on the motherboard that are swollen or leaking from the top or bottom? You would need a better PSU to compensate for those leaky capacitors, but I think it would be time to build a new rig at this point.
6. Does your PSU have the correct power connector for the GPU? or does it have enough power connectors into which you might be able to plug an adapter? Just be careful with power utilization.
7. Concerned your GPU won't fit? Try jamming it in there to find out. Just be aware of ESD (electrostatic discharge). Try to work on a non carpeted area like a table top, kitchen floor or bathroom floor. A little bit of humidity is your friend as it keeps ESD low. Your wife may look at you weird for working on your PC in the bathroom but that's ok.

Make sure PC is unplugged. Touch the case to discharge yourself before handling the GPU. Don't touch the GPU connector pins and handle the GPU by the outer edges (I am overly cautious when it comes to ESD and electronics).
I don’t know your PC knowledge, so sorry if I scared you. Bottom line is that your PSU and the availability of a driver are the biggest concerns IMO. You will be ok if you get the minimum recommended PSU for that video card. Never skimp on a PSU. Get a decent brand name PSU that is 80+ rated and has all the connectors that you need. Don’t be too concerned with the Platinum, Gold, Bronze rating levels. That has more to do with efficiency. Just get a decent 80+ rated brand name PSU for 40 dollars and stay away from the 15 dollar ebay PSU of questionable origin.