I tend to mess around with older Apple computers when I can pick one up off Craig's List inexpensively. I would say the #1 problem I run into with older Mac Pro's (2006 thru 2012 models)...is worn out power supplies. As well built as Mac Pro's are...I guess all the hours & hours & hours of operation eventually take their toll on the power supply.
When I run into a Mac Pro with a suspected power supply issue (and possibly in this case with a 2009 Mac Pro that I'm assuming previously always worked fine)...one of the first things I do is strip the Mac Pro down to as little hardware as possible.
What I mean is...run the Mac Pro with the simplest configuration possible:
* Only run one video card.
* Only run minimum amount of RAM.
* Only run one internal storage device (HD or SSD).
* Remove any other PCI cards.
* Only run one external monitor.
* No external peripherals.
* etc.
This will allow the Mac Pro to demand as little power as possible from the power supply...and possibly get the Mac Pro working again (if the power supply is the issue).
As far as testing the power supply. Since Mac Pro power supplies are not super common (relative to the overall computing world as a whole)...and since the average Mac Pro user is not the sort of user that "tinker's" with the innards of their computers (like a lot of Windows users do)...there's not a whole lot of info on the internet regarding testing Mac Pro power supplies.
To complicate things further...Mac Pro's from 2006 thru 2012 used various different power supplies...which then reduces the info available for any single Mac Pro power supply model. I've searched the internet...and there is some info out there...but it's not easy to find.
End of the day from my experience...when this sort of issue pops up...it usually is the power supply (unless the Mac Pro power button is faulty). Since there is a very high likelihood the power supply is getting weak (if it was totally bad many times you will hear a loud POP...and smell lots of nasty smelling "electrical" smoke from the power supply).
If it's a "weak" power supply...this is where stripping the Mac Pro down to as minimum a hardware configuration as possible will be a great place to start.
HTH,
Nick