Pop-up Assembly
Most ot the time when I am called out on a bathroom sink stoppage, the stoppage is almost always in the the stopper of the pop-up assembly. Usually it is hair and gunk hung up on the stopper. You can clear this simply by removing the pop-up rod and lifting the stopper for cleaning. If the body of the pop-up is heavy with gunk and sludge you can drop the trap and clean it out by pushing a rag thru the drain into your pan below. Re-assemble and test.
Replacing your pop-up will be easier with the following tips. Removing the old one will be the hardest part. Getting the old stuff off without damaging everything else in the process is where all the work is. If under your sink you have a "union p-trap" back off the nuts and drop the trap. Back off the large mounting nut all the way down and pull down the rubber gasket and metal ring. At this time you should be able to push the pop-up up and grab the flange with a pair of pliers. Twist the flange off the body of the pop-up and the pop-up will drop down. Now lets say the flange just will not come off because it is practically welded to the body of the pop-up with scale and corrosion. Not to worry, it happens all the time. I have cut the flange off from the top using a "Tiny Tim" hacksaw. You must be careful not to cut to deep on the flange because you may continue on into the sink itself. Typicall, I will cut out a small pie shaped wedge and using a slotted screwdriver, spread the flange apart to loosen it from its hold. The pop-up will drop out freely then. Re-installation will go smoothly if you clean the sink opening of any old plumbers putty and debris. Place your plumbers putty under the new sink flange and pushing the new body up, screw the body onto the sink flange. Tightening the large nut will squeeze out residual putty. Re-install your trap and complete the installation by connecting the linkage; ( lift rod, pop-up rod and stopper). Challenge the drain to test for leaks and thats a fix.