Not to discourage other answers, but this might be a better question for /r/AskAnthropology. Otzi isn't ancient, he's pre-historic, so they might well know this better than any of us here.

As it is, the oldest tattooing practices I'm familiar with are, unsurprisingly given my flair, in the near east. It became relatively common for Christian pilgrims going to Jerusalem to get a tattoo while en route in Egypt as proof of their time in the Holy Land. Traditional tattooing practices were described by Edward William Lane in his ethnographic classic An Account of the Manners and Customs of the Modern Egyptians. You can see the relevant pages if you scroll up a bit from here.

There's little reason to think that this practice or methods would have radically different in previous centuries given how simple it is. The basic method involves piercing the skin with some amount of soot or ash mixed with milk (or presumably other liquid or oil), and potentially including other common and ancient dyes like indigo to alter the coloration.

As I understand it, these basic implements, some form of ash-based ink, and simple needles are the basis for most traditional forms of tattooing, for example, in Polynesian tattooing as well.

If you're interested in primitive tattooing techniques, you can research Polynesian tattooing. It's pretty similar to modern tattooing, only the needles tend to be made from bone or ivory, and the motive force is tapping instead of a machine.

If you're interested in historical accounts of tattoos among Europeans, check out Julius Caesar's Commentaries on the Gallic War, and Ahmad ibn Fadlan's account of the Rus. According to both, tattoos were common among the Germanic and Celtic peoples they encountered, often the same blue, wood ash kind Otzi had.