After wrapping my head around your problem I could think of several reasons why your machine shuts down on it's own:

- PSU - possibly the other PSU you tried does not produce enough juice to run your machine or it has similar power outlet problem like your original one. Bare in mind that PSU's power outlet decreases over age. If both are similar age and wattage, chances are that they will both have same "aging" problems.

- the silliest reason (and, somehow most likely) is the power cable of your machine, or the socket it is plugged in. Loose connection can cause mobo/CPU dying symptom, without any error reports/BSODs whatsoever.
Simply change the socket and/or the power cable to see if that's the case.

- motherboard or CPU failure - this is worst case scenario. No real way to eliminate one or the other unless you have spare CPU (which you don't), or spare mobo (yeah, right, you sure have extra one of those lying around)

- incompatible PCI card - I had a PCI modem that would cause shutdown every time I try to dial. Maybe you have some old or no-name PCI audio card with outdated drivers?

- weird RAM issues. I experienced similar behavior of my machine back when I was experimenting with overclocking. That would happen usually when I go overboard with MHzs. Countermeasure is to increase the RAM voltage, or to under-clock the RAM itself.
What you're experiencing can be caused by RAM-motherboard incompatibility. That would be so if you haven't consulted your motherboard manual when you were purchasing the RAM. I did the same mistake and spent almost a year trying to figure out the right timings. That's what happens when you ignore the list of supported RAM models/manufacturers/configurations. I said "weird RAM issues" because one would expect memory access errors appearing in pop-up windows every now and then, but with memory sticks one can't ever be 100% certain.
Given that you have 1/2 a Gig of RAM, I take that it is one stick, all you can do with it is to change the slot.