How much work does a tattoo artist expect you to do on your own regarding the design?

You are not expected to bring anything to a tattoo studio except your I.D. Of course, if you have specific wishes for a piece that you want the artist to specifically convey, then it would be a good idea to bring a little something that at least gives a hint at what you desire. Anything else, you can fill in the blanks and let the artist know what you want. I think what you have is actually more than most can come up with. Some people just bring mediocre scribblings at best and expect an artist to make it better, which of course they can, but it's silly really. You'd be better off just using your words to describe what you want. But anyways, I think it sounds like a great piece of art you want in memory of your grandad. I would not be "pissy" if you brought that to me, and neither would any other professional artist with a proper business-like demeanor. And yes, custom artwork will cost you extra, but nothing so extreme that it will break your bank. Best of luck with your tattoo.