So, an outline of what I think the plan is/should be for Naoko's birthday:
We open with Naoko sulking in her room, thinking no one she told remembered her birthday, when she's interrupted by a knock on the door. Naoko finds Misha at the door, who asks why Naoko looks so unhappy. Naoko explains that it's her birthday, and everyone she told about it seems to have forgotten. Misha decides that since Naoko doesn't have too many clothes, and it's her birthday, they should go into the city and go shopping. Misha drags Naoko out of her room and outside, where they run into Hanzo. Misha remembers that Naoko said something about Hanzo when she had dragged Naoko into town, so she offers Hanzo the choice to come into the city with them to celebrate Naoko's birthday. (Hanzo may have been outside the girls' dorms because he was on his way to Naoko, I dunno. Why he was out there is up to you.) Together, the three of them go into the city, and Misha guides them to a clothing store, and makes Naoko pick out things she "likes" and try them on. Each time Naoko tries something on, Misha makes her show Hanzo, embarrassing Naoko greatly. Of course, this whole time, Naoko is quite embarrassed and unhappy with Misha. After Misha decides that she's tormented Naoko enough there, they purchase the clothes Naoko decided on and leave the store. After that, they walk around the city for a little bit, eventually passing a hat store. Hanzo goes into the hat store to buy the hat that Naoko wanted (black wool driver's cap), while Misha drags Naoko off somewhere else. (Maybe Misha takes Naoko to some sort of beauty parlor (<-- is that a thing? I think I made that up...) or salon or something, I really dunno here.) Afterwards, they meet up again and go to a restaurant for dinner. (Hanzo got reservations while he was waiting for the girls after getting the hat. I think.) At the restaurant, Hanzo gives Naoko the hat and gets a hug (and possibly, possibly a kiss, don't count on it though!) from her. Misha watches the proceedings with a smile, and then they all eat dinner. Afterwards, the three head back to Yamaku together, and each goes to his/her own room. (Unless Hanzo escorts Naoko back to her room, of course. Or she ends up passing out at the end and needs to be carried back to her room. That too.)
If Hanzo plays his cards just right, he could theoretically sleep with Naoko at the end of this. It's most likely not going to happen, though.