The cooling fan? Are you talking about the fan on the side of the hot-end? Or the fan in front of the hot-end. The fan on the side of the hot-end is for cooling the material as it lays down on the build plate. If that's the fan in question, it will spin up at first and then turn off for the first couple of layers. That is, unless you set it up in your slicer otherwise. The idea of the side fan not spinning during the first couple of layers is for the smoosh (if you will). If it's the front fan that spins up and then stops, you may want to look at your slicer again. Make sure it is set up to whatever you want to run at. If it is where you want it. Most likely the fan is bad and needs replaced.
As far as replacing the hot-end fan goes. It should be fairly straight forward. Soldering isn't that difficult. Just buy a decent soldering iron and solder twist the leads together and solder as needed. You can buy several materials to seal the solder job. I would recommend catching a video or two on YouTube on replacing those fans.