How to remove rotating bathroom sink drain cover
Look underneath for a large plastic hex bolt holding the drain in place.

Disconnect the P/U trap and unscrew that bolt. You will need to apply a bit of force to break the caulking. 

Putting in a new one will require also putting in some caulking. The new plug will come with instruction to that effect.

There are drains with a plug that is operated with a lever behind the tap. This may be what you are looking for as a solution.