https://www.newegg.com/Product/Combo...=Combo.3556934

Something like this combo + the GPU would cover your budget and I'm assuming you already have an SSD or HDD??

https://pcpartpicker.com/list/DZrB3F

So yea, and then if your case is ATX form factor compatible, etc then yea.
Something like that should work out. I'm not too sure exactly of everything you have but \o/

Depending on the case you have it may be easier to just get a new one for like $50
Same with PSU or SSD, and HDD

Buying a fully built computer is always going to be $100-300 more than doing it yourself from what I've seen.
But it does save you the time and hassle if something goes wrong like you get faulty products, etc so just depends on if you're willing to go through with everything building a computer requires. Which is time and patience