Turn specific weapon on auto fire without forcing all of the weapons to do so

There are three ways to avoid missiles, each with differing costs:

1. Upgrade your engines (hard to do early game, but at least keep both the engines and piloting manned to boost chances of avoiding them).
2. Cloak, if you cloak as soon as you see the shot you can avoid the missile. You'll need the ability to cloak, which you have to purchase from a shop.
3. Defense Drones (and also Mark 2 ones) will destroy incoming missiles but you'll need a drone bay for your ship if you don't start with one and also find the right drone schematic.

1 is the easiest. 2 & 3 requires finding at least one shop, and 3 requires additional luck if you don't get the right drone schematic when you purchase the drone bay. Early game, unless your ship starts out with a drone bay, you may just have to take your lumps.

If you use a teleporter to send crewmen, remember as they damage a system the enemy ship loses 1hp in hull damage as well (ruining your chance for the achievement get). Same goes for systems damaged by fire spread.

Your best bet is to get 2-3 weapons that fire missiles, and don't auto-fire them. When they are all fully charged, pause, and target drone system and weapons. The drone might be able to destroy one missile, but the others should hopefully hit.