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gjm comments on Avoiding doomsday: a "proof" of the self-indication assumption - Less Wrong
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I'm not sure about the transition from A to B; it implies that, given that you're alive, the probability of the coin having come up heads was 99%. (I'm not saying it's wrong, just that it's not immediately obvious to me.)
The rest of the steps seem fine, though.
Pr(heads|alive) / Pr(tails|alive) = {by Bayes} Pr(alive|heads) / Pr(alive|tails) = {by counting} (99/100) / (1/100) = {by arithmetic} 99, so Pr(heads|alive) = 99/100. Seems reasonable enough to me.