How should I connect this switch to my Arduino?

Depends how you want to read the switch.
NO means Normally Open (so it closes/completes/forms the circuit/counts as a button press when pushed down)
NC means Normally Closed (so it opens/disconnects/breaks a circuit when the switch is pushed down)

So if you want a standard operating button, you would just use the COM and the NO pins. Treat it like 2 pins of any other microswitch button you might've had experience with (the 4 pins you mention are just 2 sets of 2 pins). Ignore the NC pin.

So you don't NEED a resistor if you use internal pullup. So the part you say 'I know with a normal 4pin button, I need a resistor', I would check your knowledge, because it's not entirely correct.