Yes you do have to drain coolant down when replacing a water pump.
When replacing the water pump you need to drain all the coolant out of the radiator because you'll need to remove the hoses to the water pump.
The bottom radiator hose should be connected to the water pump and sometimes heaters hoses may also be connected to them as well.
So make sure to use a bucket or something and drain the radiator of all the coolant and then remove the old water pump and reinstall the new one.
Then attach the hoses back and pour coolant back into the radiator until it's full.
You'll usually have some air trapped in the coolant system but eventually it will come out as you add more coolant.