This is the only achievement that might conceivably be missable, and even this might be hard to miss. You get the achievement for clicking 1,000 times in the game. Since the "game" is not really a game, but rather a visual novel which you just watch, the clicking is just to advance the text. Whenever you click to advance the text, the next part will slowly appear on the screen. If you click before it's done it will all appear at once. This essentially doubles your click per dialog line and will easily allow you to get the achievement before the story ends.
You can do the whole game, beginning to end, in under an hour, so this is very easy.