I’ve done a few. You can definitely do it, especially with the help of YouTube and DiS.
Worth noting there are probably at least two fans in there - one on the CPU (which is the one you need thermal paste for), and another built into the power supply. There could be one (or even two) on the graphics card too, depending what you’ve got.
It’s pretty unlikely to be the one in the power supply that’s making a noise, but it’s worth checking.
If it gets worse when the computer is doing more work, then it’s more likely to be the one on the processor or graphics card. Do what @The_Cosh said first though - could save you some money.