Arduino LED backlight - should PWM be used with transistor or capacitor?
You don't need the MOSFET, you don't need the capacitor, you do need the resistor.

The pins on your Arduino can deliver up to 20mA comfortably, so in this case there is no need for the MOSFET. However, if you were to replace your backlight with a high power LED, your current setup will work just fine.

Without a capacitor, your PWM is a nice square wave. The cap will, depending on its capacity and the resistor in your circuit, turn that into a more rounded waveform, or very ripply DC. Whichever is the case, it will not benefit the (dimming of) your LED. With some bad luck it will even introduce flickering instead of preventing it!

The reason why you should still use the resistor is to prevent the LED from drawing too much current. Without it, it's simply a game between the LED and the Arduino pin to see who will burn first. Whoever wins, you'll end up with a destroyed backlight or fried Arduino. Adding a component costing a few cents can prevent that.