They both mean in one game. There are no rounds in Zombie Gunship, so the use of that word in your second achievement is a misnomer and likely a typo on their side.
I've achieved both of these, so I'm certain it's possible and within the confines of a single game up to the point of the game's termination.
The first achievement requires you to fire the 40mm 300 times without firing any other gun between shots.
The second achievement simply takes time because of the high count of 1000, and, since there's no in-game counter, it can be difficult to judge how many you've unloaded. A single overheat will at the very least reset your count, perhaps even disqualify that game.
For the second one, just make sure you're constantly firing the 25mm right from the start of the game... even while you're moving around looking for zombies. Fully upgrading the cooling and rate of fire for that gun obviously helps with this achievement.