Use a small flat screwdriver to lift that pop-up assembly. That pop-up is either resting on top of the rod or it is threaded through the pop-up rod. If it doesn't remove easily than most likely the rod is holding it in place. Adjustment of the linkage can sometimes correct the problems associated with pop-up drain assemblies.
More than likely you will find nasty hair and gunk that has built up over time.
What could be your situation is that the rod either rotted off that actuates the pop-up assembly or there is limited movement in the swivel of the pop-up rod.
I would say just see if the plug will come out; inspect/clean and see what is actually wrong with the mechanism.