180 ohm resistors are hard to find? Where are you looking?
The MOSFET does not convert any voltage - it merely turns on and connects the LED cathode to Gnd. The LED anodes are at +12 already, so closing the loop electrically lets the LED strip turn on.
MOSFETs are voltage driven devices - a Logic Level N-channel MOSFET needs ~4.5V to turn on fully. The Gate has some capacitance associated with it - we use the 30mA out of the Arduino to charge that capacitor up quickly (and discharge) so the MOSFET turns full on & full off quickly.
The cap looks like a short to ground when discharged - the 180 merely limits the current that charges the cap.
5V/30mA = 167 ohm; 180 is a standard value.
A N-channel MOSFET, Logic Level, Low Rds, fairly low gate capacitance, thru-hole, part like this would work well