Can I use caster sugar instead of cane sugar?
Yes you can, so long as you work by weight not volume. Unrefined cane sugar (or refined sugar with some molasses added back in) would contribute a little to the flavour, but sugar mainly provides sweetness, and in brewed ginger beer, a source of nutrients for the culture.

Caster at the same volume as granulated would be too sweet (the finer crystals mean more will pack into the same space), but at the same weight it would be fine.