First of all, don't get the cartridge from Davis. They charge $20 plus $40 shipping to Hawaii!!! I got one from Scaled Instruments
www.scaledinstruments.com for $16.50 plus $3.75 mailing, so in total I paid only 25¢ more than Davis's normal price.
It is necessary to remove the cups, as they will have to go on the new cartridge. There is a set screw on the cups' body just below the overhang of the cartridge housing that requires a 0.05" allen wrench. Be careful; it's easy to lose that allen wrench!
There is a screw on the cartridge housing part way back on the mounting arm that must be completely removed. It screws into the upper half of the anemometer/weather vane housing and holds the two halves together. Use a #2 phillips screwdriver to remove the screw. Once the screw (which unfortunately is not stainless steel) is removed and safely stowed, you can pull the anemometer wind speed cartridge down from the upper half. It does not slide out. It comes straight down.
Then simply push the new anemometer wind speed cartridge into place where the original was and insert and tighten the screw.
Push the cups back onto the shaft all the way and tighten the setscrew with the 0.05" allen wrench that you carefully did not lose, and you are all done except for safely climbing back down.