So I can go from n = number of rolls to a confidence interval by way of simulation, but (to answer the original question), I've got no idea how to go from a confidence interval to a number of rolls by calculation.
Pretty reassuring that my method and yours both produce an answer of about ~300 for 95% confidence!
I think for an analytical answer it's going to some sort of Bayesian thing; a posterior probability distribution that allows you to calculate a confidence interval and a uniform prior distribution?
So I can go from n = number of rolls to a confidence interval by way of simulation, but (to answer the original question), I've got no idea how to go from a confidence interval to a number of rolls by calculation.
Never enjoyed stats anyway.