Are you new at business? Just curious. Shouldn't you already know this stuff? Well, we all have to learn this stuff the first time I guess.
I'm assuming you want him/her to run the day to day stuff.
Like any store manager, he or she would need to no how to keep the inventory and order new inventory (hopefully find better deals to save money as well). Needs to know how to fill out payroll details and possibly local and state tax information if need be.
For tattoo shop managers: how to maintain a clean and sanitary shop and keep the shop up to or higher to the standards of the local and state health codes that apply to tattoo shops.
Needs to know how to market the shop in order to drive in new business. Needs to keep the shop to a certain standard in order to keep new customers coming back. That might mean maintain communications with the shop's best customers. Give them incentives for bringing in their friends through word of mouth as well.