Fill the cup 2/3 full with ice. Pour in half to 2/3 of the ginger beer.
Then, add Smirnoff Vodka, the coffee liqueur, and squeeze in the orange juice. Give that one or two good stirs, then fill the rest of the way with the remaining ginger beer.
Garnish with a cinnamon stick and another orange wedge or a piece of the rind.
Notes
The copper mugs I own can hold 16 oz. of liquid (this is if you do not add ice, just liquid). If you do not use this size of cup or glass, you will likely need to adjust the above recipe to fit your needs.
I used a jigger to measure the vodka and coffee liqueur here, as mine has two cups with one measuring 1 oz and the other 1.5 oz. If you use a shot glass to measure instead, be aware that usually a shot is 1.5 oz. So a bit more than you need here.
This would be fine for the vodka, but on the coffee liqueur, I’d be sure to use a scant amount in a shot glass. Otherwise, the flavor could overpower everything else, and we’re aiming for balance here.