It sounds like your old system had the thermal paste fail. If it's not been reapplied for 9years, that's likely where the heat issue is (And maybe a clogged heatsink too) Thermal paste, on average, is only ideally good for about 4 years.
Arctic silver 5 and other high end pastes might last a bit longer, but generic pastes I don't think have a good shelf life to them.
But anyway, for that Acer system you listed, there's 2 things to keep in mind:
1: It's a small form factor, that's mATX at best, so you aren't going to be fitting most video cards in there unless their small profile, single slot cards, like entry level, usually older cards.
2: The power supply in it is only 220watts, it may not be enough to power an external video card, esp depending on which yours is. You'll only have maybe 60watts to play with over the systems needed power (assuming you use 3 USB ports by my calculations), which likely means no 6pin connector, it will have to be run off the PCIe's power only.
Some lower end cards can fit that in both size and power, but you will not have much power to play with, and almost zero power to account for capacitor aging in the power supply.
What video card are you thinking of putting in if you buy the Acer, I could see if it will have enough power and fit^^
Last thing to consider is, unless the video card uses passive cooling, that tower will generate a fair bit more heat that it isn't designed to deal with.
All things considered however, that would be plenty to run FFXI very well.
Another note is you could build you one, if you feel comfortable doing so, or have the experience, AMD makes it super possible now with their APU's, and if you reuse parts and case from your old PC, you could build a pretty decent gaming system for less then 500$. It won't be steller like an i5 system with a geforce 700 card like you might get at 800$, but you'll get more then plenty of power for gaming, far beyond what FFXI needs.
The less you can reuse, the slightly lower end the system will be however, and the lower price you have to go for parts, the less feasible it will be to build your own, and might save money buying something cheaper like this Acer you found.
If you wanted, I could do some searching for pre-built systems, or write up a wishlist for a 500$ system to build yourself^^