Did Asako and Souhei end up being together in the end? I have watched the anime and read the manga, but the ending of the manga didn't explain if Natsume Asako and Sasahara Souhei end up together.

Did I miss something, or is it just not mentioned?
On Natsume's work desk, in the manga series credits, there is a picture of her smiling and hugging a guy with a very similar hair-style to Sasahara's (the picture is actually the same on the back cover of the volume 13, after removing the jacket). This suggests that they indeed did develop a further relationship after high school. 

At their friends' wedding, they are always seen with each other, though it is not confirmed if they are a couple. So it's up to you to decide, I guess! Personally, I think that they are too great couple to not be together :)