Here's the thing: in Bayesian updating, probability mass moves from one hypothesis to another. That means you need another hypothesis. For Bayes' theorem, you need three terms: the prior probability of your hypothesis (99%), the probability of seeing the experimental result assuming your hypothesis is true (0.0001%), and the probability of seeing the experimental result whether or not the hypothesis is true. You need another piece of information, and that will depend on your state of knowledge. For instance, the coin could be perfectly fair, slightly balanced in favor of tails (40:60), slightly balanced in favor of heads (60:40), 90:10 (either direction), etc. You need another hypothesis, or probably a bunch of them.

If you want to make the math simple, you can just consider two possibilities: your hypothesis, and the null hypothesis (the coin is perfectly fair, 50:50). If that's the case:

P(A|X) = P(A)P(X|A) / (P(X|A)P(A) + P(X|N)*P(N))

P(A|X) = (.99)*(.000001) / ((.99)*(.000001) + (.5)*(.01))

which works out to: P(A|X) = 0.000197961... (Nota bene: Check my math. I haven't slept much recently, and I could easily have made any number of mistakes)

Again, this is a really simplistic calculation, because there are so many other plausible hypotheses, and I haven't studied quite enough probability theory to do all the calculations, even if I had more information.