I used to talk with my neighbors a lot, but these days i’m very busy with my company and I didn’t have a chance to talk with them.

-> I used to talk with my neighbors a lot but these days, I’m very busy with my company and I didn’t have a chance to talk with them. (Use capital letter for your subject I.)

however, yesterday I ran into my upstairs neighbor in the elevator when I was on my way to work! I was very glad to see her in the elevator and we said hello to each other. after that, we talked about the weather first because it was getting so hot.

-> However, yesterday I ran into my neighbor upstairs in the elevator when I was on my way to work! I was very glad to see her in the elevator and we said hello to each other. After that, we talked about the weather first because it was getting so hot. (Take note of your word order and start with capital A since it's the beginning of a new sentence.)

after getting out of the elevator, we walked to the bus stop together.

-> After getting out of the elevator, we walked to the bus stop together. (Use capital A since it's the beginning of a new sentence.)

she asked me about my company life and I said I was really busy with new project. also, I asked her about her job. and she said she is teaching at highschool.

-> She asked me about my work life and I said I was really busy with a new project. Also, I asked her about her job and she said she is teaching in highschool. (Use capital S and you can substitute "work" instead of "company".  Start your new sentence with capital A and your preposition should be IN for highschool.)

because my wife is teacher also, I was happy that I could connect her with my wife.

-> My wife is also a teacher and I was happy that I could connect her with my wife. (Start your sentence with capital M and take note of your word order.)

it was so fun to talk with her and I lost track of time.

-> It was so fun to talk with her and I lost track of time. (Start your sentence with capital I.)