Faucets in a pop up depend on what type of system you have. Older units usually have the pump style while newer models will have a water pump, water heater and standard faucet.
After our pump faucet showed leakage I replaced it. That was pretty easy but since the leak rotted the counter top the screws wouldn't hold. I ended up cutting a thin piece of white masonite to glue to the counter where the faucet was, and use long bolts and washers underneath to hold that one in place.
After a year that one showed signs of leaking, so I went another route to fix that.
I bought a water pump off eBay for $12, an on demand type. It will run when the faucet is opened, and shut off when you turn the water off. I hooked it up in the line between the water tank and the faucet. Used wire clips to wire it in, and added an on/off switch for those "just in case" moments. I used a standard bathroom faucet, a cheap one that the height was short enough to allow the counter top to fold over. The counter top was re-done at the same time due to rot from leaks.
Another job I tackled while I was working on the counter top was the drain. The trap used in the original design had a metal bolt to hold the trap to the tail piece. The water drained about 1 1/2 inches above the base, so the bolt was always sitting in water. Not only was it rusted, but it blocked lots of gunk. I glued in an adapter to hook up straight to the drain line, eliminating the trap. It now drains as fast as water goes in the sink.