You do NOT have to remove the power supply from the computer case, but you must unplug it from the board.
If your motherboard as the 24 pin socket, you must have a power supply tester that also has 24 pins.
If your board has 20 pin socket, and you have a 24 pin tester, you can use the first 20 sockets.
It is easier and safer to remove and test the power supply out of the case.
The user manuals are insignificant extras, when available. Most power supply testers do not have a manual. You plug them in. They work by showing the results on LED's or an LED screen.
Five minutes is plenty long enough to test the unit...
The better, more expensive power supply testers will also test the power supply under load, but you are talking about $75 to $100 bucks.