There are some different approaches you can take, but they all have risks.
The screw is broken off a little bit below the top of the post. Drilling could be one method, but again it risks drilling out the threads if you are out of center by even a small amount. Two other approaches could be used:
1 - Use a screw head file, and cut down through the top of the post far enough to create a slot in the remainder of the screw, then use a screwdriver to turn the screw fragment out.
2 - Use a watchmaker's lathe to cut away the top of the post, exposing enough of the screw to get ahold of it, so it can be turned out.
Two problems with these methods really, and one is that if this is a modern watch, the screw will be held in place using some Loctite, so unscrewing it is going to need some torque - heat can help break the bond but still the screw fragment may be difficult to remove.
The second is that the post that the screw threads into is a sealing surface inside the pusher. So the O-ring inside the pusher tube seals on this post, so if you remove too much of the post, that seal might be lost.
By the way, you don't want a jeweller for this, you want a watchmaker - very different things.
Cheers, Al