R = U / I
R = 5V - 3.2V / 0.02A (Each led is 3.2V and takes 20mA)
R = 90 Ohm --> Is 100 Ohm resistor before each LED right?
Will it work, or will I damage my Arduino?
No...that calculation is for a 1.8V LED. The voltage isn't 5.0-3.2, it's 3.2.
R = 3.2V / 0.02A (Each led is 3.2V and takes 20mA)
R = 160 Ohm, minimum.
nb. That resistor value is the minimum. In practice you should use something a little bit higher than that in case one of your LEDs is only 3.1V (the datasheet will tell you the tolerances).
Your circuit doesn't need a resistor on the GND wire. Apart from that it's fine.