Yep,
That looks like a Walbro WYL (and you really don't want any other WYL than the one made for an EH035).
From the looks of it from the outside, it likely needs nothing more than a simple disassemble and cleaning. There are a number of very small delicate parts inside this carb, so it is a job for a clean workbench and a clear head. With a little searching you should be able to find a diagram of the carb online which would be very helpful for a first timer. If you are not sure of your ability with this project, a local mower repair shop would certainly be able to handle it.
The appearance of the primer bulb (being dark and opaque) strongly suggests that the engine has been sitting for quite a while with old gas in it. I would definitely replace the primer bulb and the lines too. Also, clean out the tank and check the filter/pickup as well.