PC keeps shutting down unexpectedly

Can anyone help me out?
I'm dealing with a PC that has these specs:
Processor: Intel(R) Core(TM) i3 CPU 540 @ 3.07GHz
RAM: 4 GB
32-bit system
Graphics: GeForce GT220

The thing just shuts itself down whenever I try to play games or even just browse the web...
It was actually in the shop for repairs about two months ago.
Got a new Seagate installed.
Power supply is 385W.

Is it shutting down completely or just restarting? It could be a faulty RAM stick, but honestly, it might just be a buildup of dust inside the case.

You should probably grab MemTest86—either run it directly or throw it onto a bootable USB—to scan everything and see if any errors pop up.

Also, go ahead and download HWMonitor. Once it's installed, just paste a screenshot of your temperatures here so we can take a look...

http://prntscr.com/5pjbf5
Sometimes the whole thing just dies completely, and other times it restarts, but then my monitor, mouse, and keyboard won't even respond...
Every now and then, it doesn't even have enough juice to turn back on.

Maybe run some kind of game with HWMonitor open in the background, then just grab a screenshot after you've been playing for about five minutes.

Also, what are we working with for a power supply? What’s the actual brand and model on the sticker? It might just be underpowered, I guess.

http://prntscr.com/5pji0q

I can't really get to my PC right now to check the exact power supply model.
But I'm pretty sure it's a 385W unit.

The temps aren't even the issue here; the problem is definitely somewhere else.

My money is on the PSU. You mentioned it struggles to reboot, which is a huge red flag. Plus, those 12V rail readings in your screenshot look pretty low, though I guess software sensors can be total liars sometimes. Maybe try swapping in a different power supply if you can borrow one? In the meantime, you could always crack the old one open to blow out the dust and see if any capacitors look like they're about to pop.