"I walked down the back alley of Yokohama." You don't really walk down THE back alley of a city, because there are multiple ones (unless there IS only one back alley and I need to study geography better, in which case you can disregard this. If I am correct then:) change it to "I walked down a back alley in  Yokohama." and maybe elaborate.

"A little off, there were those colorful small restaurants that sold ramen. Like, Ichibandoshi ramen shop, which was infamous for its takeouts." There is a comma splice after "like". Take out the comma. It is unnecessary. Also, capitalize "Ramen" and "Shop" because they are both part of the ramen shop name, right? Therefore they are both proper nouns. Another issue is that "infamous" means well known for some evil or bad deed, which I don't think you want to convey. Change it to a synonym of "famous". Next, the sentence is a dependent sentence (it's not complete), so actually combine it with the previous sentence and form a complex sentence. Lastly, the first sentence's adjectives should be switched around for easier sentence flow. The sentence, once changed, should look like this: "A little off, there were those small colorful restaurants that sold ramen, such as the Ichibandoshi Ramen Shop, which was renowned for its takeouts."

"'Oh, kon’nichiwa!'A man greeted me, and bowed." There's a comma splice again and you don't need to capitalize "a". "Oh, kon'nichiwa!" a man greeted me and bowed."

"'Ne.' and I leaned against the counter." You don't need the 'and', and make the period after 'ne' a comma. "'Ne,'" I leaned against the counter."

"But I was content to think of him. At least, dream of him." It seems like you're trying to emphasize 'think of him'. also, the second sentence is a dependent sentence and should be combined with the first. "But I was content with dreaming about him and even just thinking about him. "

"Walking back, the streets were all lit up, since it got dark pretty fast." Comma splice. "Walking back, the streets were all lit up since it got dark pretty fast. "

"Cars zooming in and out on the main highways, lights going on and off on signboards, windows filled with sale and discount coupons… the Christmas vibe." Dependent clause again, wrong prepositions, and could be better by using synonyms. Also, don't use elipses (...) like that. "Cars were zooming in and out of the main highways, lights were flickering on and off of signboards and windows were filled with sale and discount coupons— this was the Christmas vibe. "

"It’s all I’ve been doing for the past 11 months, idolizing, maybe even worshipping him." Use a semicolon. "It's all I've been doing for the past 11 months; idolizing, maybe even worshipping him. "

"I can’t remember how my life was before he came in." Add a direct/indirect object. "I can't remember how my life was before he came into it. "

"On the way I was walking, I got a glance into a music shop." You don't "get a glance" into a music shop and you don't need "I was walking". "On the way, I glanced into a music shop."

"As usual, I walked near the railing, the very edge of the space that was guarded by iron railings and leaned my elbows onto it." add a comma after "railings" and you don't say "leaned my elbows onto it". "As usual, I walked near the railing, the very edge of the space that was guarded by iron railings, and leaned forward onto it with my elbows."

"Just 781 miles away, he was sitting in his dorm or his house, relaxed. 781 miles away, he was probably thinking about how he'd spend the day tomorrow. What was he doing now?" you have to decide on which tense you're using. You keep on switching between present and past. 

"and oh so badly I want to see him." it rolls of your tongue nicer if you say: "I want to see him oh-so-badly."