According to who has already researched this: "Although no astronaut to date has had any tattoos during their employment with NASA. NASA currently has no rules against astronauts having visible tattoos.
But i still don't think that answers the question, because there is more to it.
It all depends on two things; What type of astronaut do you want to be? (pilot, mission specialist, commander, etc), and how do you plan on getting there?
To be a pilot or mission specialist, you will need so many hours of "in command" jet piloting. To get these hours you can join the navy, air force, or a private flight program.
Recently the navy has slackened their rules regarding visible tattoos.
The Air Force allows a tattoo to cover up to 25% of the body part shown in uniform.
The navy, I believe, will allow 25% of your arm to be covered. so no sleeves and nothing past like 2 inches before your wrist.
Different flight schools have different rules and regulations regarding eligibility. Contact the school your interested in to learn more.
I myself want to be an astronaut. Once i get my bachelors degree in aerospace engineering, i plan on joining the air force. I hope by then they will have slackened their rules, because i have a sleeve tattoo on my left forearm. Next week i have an appointment to get a quote on laser removal."