what's the least painful way to replace faucet cartridge?

Water off under the sink, open the tap to drain it. Pop the cap off the handle, undo the screw, pull the handle, then remove the retaining clip or nut over the cartridge. Before you pull the old one out, note which way it faces - put a new one in rotated and your hot and cold swap. Take the old cartridge to the shop with you, there are loads of nearly identical ones and guessing is how you end up going back twice. If it's seized, a cartridge puller is cheap and saves damaging the tap body.