From what I can see there, which admittedly isn't alot, there are two silver traces. What most likely happens is the bottom of the plastic button has a conductor on it and when you push them down, it connects the two. If that's the case, you need to figure out which one is ground (should be able to use a multimeter and find out which of the two has connectivity to the same trace on another button) and then wire all those to the ground of the arcade buttons and the other trade to the NO pin on the buttons.
Then again, a clearer pic might totally prove me wrong.