Use a small suction cup on it or a utility knife to pry it up from the rim.
Now that you have added the additional photo and comment that there is no access behind the wall (assuming it is drywall which would have to be cut out and replaced after repairs)
I suggest: When you do remove the plug, instead of using that type of plug, use a bathroom sink strainer to prevent items from going down the drain on a daily bases and when you need to stop up the sink use a rubber stopper!
This website has a great animation on the inner workings of this mechanism.
You didn't state whether the back lever can move up and down freely??Put your fingers on the top of the plug and try twisting the plug around if you can. Some of them have a locking mechanism in them that can get stuck on the pivot rod.
Most likely what happened: