Yes you can, Infact you can do it even if the earlier Owner got that achievement, aslong as its a different Live account then it should be a new achievement for you.
Yes, me and my boyfriend rarely by games new, as it's so much cheaper to get pre owned. Your achievements are stored on your hard drive and not on the disc, so no matter how many people have played the game on different gamer tags, you'll still will get the achievements.