Seems to me like you'd want to go first... because at most, you get 3 attempts... and by going first there is always a chance you don't die on each one of your attempts, whereas by starting 2nd, you end up at 100% chance of death on your 3rd try... plus your odds are technically better for each one of your attempts if you are 1st...
1st attempt: Player A 1/6 die, Player B 2/6 die
2nd attempt: Player A 3/6 die, Player B 4/6 die
3rd attempt: Player A 5/6 die, Player B 6/6 die
Although, I'm not exactly sure that is a mathematically sound explanation... perhaps it is more of a "gut instinct" answer??
However, the reality is that for player B to shoot, it requires player A not to die. That would change the probability.
For example, for the first attempt:
There are
Player A dies: 1/6 chance
Player A lives, Player B lives: 5/9 chance
Player A lives, Player B dies: 5/18 chance
Following this, what I'd previously posted begins to flip the odds in B's favor (provided my math is correct) quickly. Right now it seems as if Player A is the better option: and it would be, if it was only one round.