Resistor will be small, like 180 ohm.
Just prevents too big of an inrush of current into the MOSFET's gate capacitance.
Say the cap was a dead short to ground - you want to limit the arduino's output current to say 30mA.
V=IR, or V/I = R
5Voutput/30mA = R = 167 ohn, so 180 will do nicely.
I'm assuming your LED strips have current limit resistors built in, so I didn't add any to the diagram.