Corrections in red.  Stylistics, comments, and variants in blue.

One Rainy Morning  In a title, capitalize the first, last, and all other words, except:  articles, prepositions, coordinatingn conjunctions, the word "to" when it is part of the infinitive of a verb, and the conjugated forms of "to be" (am, is, etc)

A typical rainy April morning on my way to the office. It takes me half an hour to get there. Always in a hurry, trying to manage everything in time. Not to miss anything. Not to be late.
But here it is, all of a sudden - a 5-minute-stop. A closed railway crossing on my way. All the cars stop, waiting impatiently for the train to pass. Somebody smokes nervously. Somebody pulls/turns his car around, not willing to wait.
I'm sitting in my car, looking out of the window, listening to Sade's "Smooth Operator," [1] and enjoying this short moment of calmness/calm. It's raining heavily, and I can see the rain drops on the window glass, and there is the river on my left. I have just noticed it is free from ice already. Everything stops for a moment - just rain, river, and music: Time stops....
But now the railway crossing gate is open, and the cars begin to move fastly. [2]
I am on my way to the office. Not to be late. Time flies again.

[1] Names of artistic works are italized.

[2] "fast" also serves as an adverb

Your post is written extremely well.  It sounds very natural.