Use transistor to press button on external device
Yes, you have screwed the circuit up slightly.

You want to lose the red wire between the collector and +5V otherwise when you turn the output pin HIGH the transistor will explode.

Also you need to make sure that the ground of the Arduino (emitter of the transistor) is connected to the - of the target circuit and that the button you are "pressing" is an active-low system with internal pullup. If it's not then you will have to come up with another way.

An opto-isolator is a better choice than a transistor when you don't know the internal circuitry of the target device: