I did not say you have to light 4 LEDs at once, I said up to 4 LEDs. If you choose to light only one, you can do that. But if you truly only want only one led to appear to be lit at once, you do not need to multiplex
120R is probably ok, but do not go any lower than that. The current into your leds is probably around 20mA, which is the maximum for most LEDs. In saying that, I am assuming the ULN chip will drop around 0.7V and the led's forward voltage is around 1.8V, so the current is ( 5 - 0.7 - 1.8 ) / 120.