MTG Doomsday card, cast at 1 life
First of all, Doomsday doesn't deal damage (so it won't be stopped by effects that prevent damage). It causes life loss.

Secondly, the amount of life you have when you cast Doomsday is of no relevance; it's the amount of life you have when you are instructed to lose life when it resolves that matters. 

Half of one rounded up is one, so you drop to zero life if you have one life when the spell resolves. If you still have non-positive life the next time a player would get priority (probably right after the spell resolves), you will lose then.