Hooking up to a contact already in a circuit board is tricky. You have to be careful to not blow up your Arduino (too high voltages can do this), but also to not have your Arduino appear to the existing circuit as closing the switch. Less likely to happen, but still possible.
Detecting a button state change for a period of time is easy to accomplish.
If you want to use a WiFi connection, you may be better off with an ESP8266 based board, such as NodeMCU or WeMOS D1. A pretty powerful microprocessor with WiFi built in already. Much easier to work with than Arduino + WiFi shield (which may be built around an ESP8266 anyway).