Latisse should not have any affect on an eyebrow tattoo. Latisse is gently applied along the base of the upper eyelashes and affects only the hair follicle. Eyebrow tattoos will fade over time anyway but this will not be changed by the Latisse.
Latisse takes time to work. Only use it once a day. Use of Latisse more than once a day will not increase the growth of eyelashes more than use once a day. As long as you consistently apply it at the same time each day, whether it is in the morning or at night should not matter.