I've made some corrections and provided a revision below. I did not correct the sentence fragments because I think you were going for something more poetic. Fragments for stylistic reasons are fine as long as there is some meaning or imagery conveyed.

Here are some corrections:

Spring! Finally, the spring came to my city. We have all been waiting for this for such a long time. Three months of autumn, three months of winter. Wind, rain, gray and monotonus city, and then cold and the mud again. When I look out the window, it seems like people don`t live, they survive, and have turned into zombies.

Here is a revision (with fragments):

Spring! Spring has finally come to my city. We have all been waiting for this for such a long time. Three months of autumn and three months of winter. Wind, rain, and gray all over the city, and then, cold and mud. When I look out the window, it seems as if people are not living, but merely surviving. They have all turned into monotonous zombies roaming about the streets.

But everything changed a few days ago! The sun shone on the pure, blue sky, returning smiles and joy to the people. They have awoken from their hibernation and have started live, rejoice, and love. If only you knew how much I love spring! There is so much hope in this one word.