Surely it's best to go second! Whether you go first, second, third, fourth, etc, when the gun is in your hand, the chance of the shot being a killer is exactly the same. But...

During the first go, the shooter has a 1/6 chance of dying but the opponent (the second shooter), who doesn't have the gun, has a 0 chance of dying.. because they don't have the gun! Therefore your chance of survival is massively greater if you go second!