In short, you win and live to tell the tale. YAY!
If it bleeds, we can kill it wrote:
You can't die until the end of your next turn
(If an effect would kill you, ignore it. After your next turn, if you have fatal damage you die.)
The card says "You can't die until the end of your next turn" and the extra text it specifically says "If an effect would kill you, ignore it." So you most definitely do not die THIS turn from "Three hundred city blocks" (or anything else for that matter).
At the end of THIS turn, the predator has been killed and you are alive. Any concerns about NEXT turn cease to matter. Sure you were on the brink of death and might have still died from your wounds if you had to sit around for just a bit longer, but you survive and are presumably extracted just in time because the game is done.
This completely ties in thematically too because that is exactly how the movie ends, with the fatally wounded predator setting off his bomb and Arnie being rescued having barely survived the explosion.