I have the same case (Aopen H340D) in black. Right now it's sitting unused. Had heat problems also. The power supply seemed okay, but the fan hardly moved any air. It isn't "standard" mATX, so I think you're outta luck for replacements. My power supply ran REALLY hot. I had a good HSF, but hot air wasn't being exhausted from the case. It would run fine and quiet when not overheated though...
I'd say definately try the suggestions first - RAM from other machine, case off with house fan pointed in, hooked up to known good power supply. No, wait, if you DON'T try the suggestions first, take up another hobby. Not doing these tests would be a disservice to whoever has to pay the $30 you say is the dollar limit. Unless you have a time constraint (which doesn't go with a money constraint), throwing new hardware at a problem without narrowing it down is just throwing money away. If it still isn't stable with case off, different RAM and different power supply, then likely culprits are motherboard (you didn't specify) and CPU (I've had a few tbirds needing better cooling over time, as if it deteriorated). If it IS stable, then narrow down by swapping back original RAM, then original power supply to narrow down the cause.
These forums are great for suggestions, but YOU have to do the real testing because all the suggestions are from only the limited information you've given us. Good luck on this endeavor and get back to us with the results of your testing.