Cosmetic tattoos are still tattoos, and they involve having the skin punctured by a needle. There will necessarily be some pain involved.
However, your skin will be numbed by a local anaesthetic before and perhaps during the application. This should be quite enough to take away most of the pain involved. Many people say that the sensation merely tickles because of the topical anaesthetic.
You will likely experience some swelling and soreness for several days after the procedure. An over the counter painkiller will be all you need to deal with this pain. Ask your artist what he or she recommends and how long they think you will need to take it.