Playing through on Normal for the first time, I wasn't able to get this achievement. No matter how careful or fast I tried to be, there was enough collateral damage to keep me from earning any better than a B ranking on the first rescue mission.
(Update: Method 1)
I found that this is pretty easy to get a bit farther into the game, on the fourth rescue mission. There will be an alert over the City 2 area after ADA is infected with a virus. This will leave you with no health reserves and you'll die from one hit. However, there are just three squads to defeat in City 2, and if you take the offensive, they're not hard too wipe out. Do charged attacks and rush-dodge a lot.
(Original: Method 2)
Others have posted tips for the Damage Control achievement, which is for getting an A Rank in all 5 rescue missions. This solution should help you at least get started. The Good Samaritan achievement is for an A Rank in any one rescue mission.
To get this, I started a new game on Easy. I rushed through the beginning section, skipping cut scenes. It only takes a few minutes to arrive at Factory 1. After defeating all the enemy squads, leave the area via the Start menu, and then enter it again. The squads re-spawn, and you can earn additional experience by taking them out over and over (leave the area, then come back).
Your experience progress can be seen as a white bar under your health bar, at the lower left. I was able to "grind" up to level 5 in about half an hour this way, although it takes more time between 3-4 and then 4-5.
If you're doing this while you're farther along on another save, be sure not to save over your main save.
At level 5, leave the area and enter Town 1. Defeat the boss and then the enemy squads. Leave the area and immediately save. This is the point where you have the optional Rescue Mission in Town 2. Watch or skip the explanations, fly toward Town 2, and enter.
Using a tip from a video in the Damage Control achievement, I went for the enemies at the far left first. Get over there fast by dashing with RT. Then lock on (you can lock on with LT as soon as you get close enough that the box around them begins to flash a little), then back up to draw them away from buildings. Stop, hold RT, then X to blast them with an energy ball. If you're in close, stop, hold RT, and X to do a devastating spin slash.
Next, go for the enemies on the far right. You should be trying to lock on and draw them away just a little, but don't don't take too long. Focus on the raptors (the things that *don't* look like big church bells) first. Try not to change your lock-on, and do one enemy at a time.
When the achievement unlocked for me, I had something like 8.6% damage to buildings, and 100% civilians saved. If I'm correct, anything worse than that will get you a B or a C. For instance, while playing on Normal on my main save, I had something like 18% damage to buildings but 100% civilians saved, and still got a B. I would recommend exiting to the title and re-loading your save to attempt it again if there are any civilian casualties (you can hear it announced, or watch the stats at the upper-right), or if you hear more than one or two announcements that the surroundings have been hit.
The key, for me, was playing it on Easy and grinding to level 5 ahead of time. Because once you're presented with the Rescue mission, it's kind of too late. If you keep re-entering an area to grind, there will be civilian casualties before you even start the rescue mission.
Good luck!