The elements of 2Z are ..., -6, -4, -2, 0, 2, 4, 6, 8, ...
The elements of 8Z are ..., -42, -16, -8, -0, 8, 16, 24, ...
The elements of 2Z/8Z are 0, 2, 4, 6.
You can consider these as being 2x the elements in Z_4, i.e. 2 x {0, 1, 2, 3} and do all the work on Modulo 4 arithmetic and multiply everything by 2 when you've finished, or you can directly use modulo 8 arithmetic and create the multiplication and addition tables based on the defined subring of that.
As long as you know how to do modulo n arithmetic you're home.