120 would work okay too and is a standard value.
The ULN2803 can be thought of as 8 transistors each with current limiting base resistors in a easy to work with DIP package.
So instead of having 8 transistors and 8 resistors, you just have 1 DIP package.
The Arduino can drive its input directly.
The outputs can accept up to 50V and can sink several hundred mA of current.
In this case, 5V and up to 160mA are well within its capability.