In addition to the above:
Bring a (supportive) friend. They can distract you and help you keep your mind off what's going on. Your thinking about the needles et al is the problem, so keeping your mind off of it helps. Even if you don't talk & just pay attention, it will help.
You won't see the needle unless you want to, and ditto for the blood. Depending on the location, in fact, you might have to go out of your way to see anything.
Although a needle is used, it's not often visible as such, and the sensation is more like scratching a sunburn or getting an electric shock than anything else. I *hate* getting shots, but tattoos are just annoying in sensation.
Blood tends to show up in limited amounts, although for redheads it can be a bit more.
I'd say let the artist know your concerns. They've likely dealt with this many times before, and will be fine with you taking breaks, getting a sip of ice water (helps some folks), perhaps distracting you with their own talk, and so on.
Fainting actually isn't that common, especially when you follow the above and the other advice given. Best of luck!