May I suggest that the real 'agony' is in the waiting, tension and anticipation - not the actual event? I was going to suggest some benzocaine containing anaesthetic skin cream, but on second thoughts, it might possibly have adverse effects on the tattooing process, make the ink run or something - I have no tattoo experience, so that's just a precautionary guess?
I would suggest you work on the psychology of this, rather than the pharmaccology. True, benzos will do you no harm in reasonable doses, as a one off remedy to overcome your dread, but IMHO it would be better (and far more 'satisfying' perhaps?) to avoid the dread if you can.
There are many ways of doing this. I mean, you could have a general anaesthetic, treat it as an operation, but that is patently absurd, unnecessary and (to be rather mean) cowardly? My personal solution would be to use that well tried, very traditional method that is not only the anaesthatising remedy for the pain, but the very root cause of many tatoos - i.e. the best part of a bottle of Captain Morgan rum... but of course, no decent, responsible tatooist nowadays will touch any customer who staggers in reeking of booze and singing sea shanties! Tranquillisers in small doses are really just the modern, scentless equivalent in a way?
What I'd suggest, if it is posible, is that you employ the help of a couple of trusted friends, get them to explain your problem to the tatooist, who will no doubt understand if worth their fee, and at some unexpected time, suddenly get picked up by surprise, delivered, and before you have time to panic, it will be over. This is the way amputations and dentistry were done with no anaesthetic in the good 'ole days, and though I appreciate you don't want a work of art rushed likethe surgeon's knife, once underway I think you'll be fine - and proud of yourself!
Whichever way you do it, very best wishes and good luck. And don't forget, it is a thing of beauty you actually want - not some torture you are obliged to endure!