Pour most of the apple cider (juice) into a wide mouth gallon jar, leaving several inches of head space (you will have a few cups of unused cider left over).
Add the ginger slices, sugar, and ginger bug to the cider and stir well to dissolve the sugar.
Cover the jar with a cheesecloth and rubberband, and put in a cool and dark place to ferment, stirring vigorously daily with a wooden spoon.
After a day or two you should start to see some active fermentation. Let it ferment for 5-7 days, then strain with a fine mesh strainer and transfer to flip top bottles using a small funnel.
Let the bottles sit at room temperature for a day or two to build up carbonation, then refrigerate.
Be extra careful not to bottle too soon so that you don’t get exploding bottles. If fermentation hasn’t died down after a week I would wait a little bit longer before you transfer to bottles.
Apple ginger beer does contain alcohol, about equivalent to a normal beer. If you want it a little stronger or sweeter, add a little bit more sugar.