|A very well written short story. Brings in a decent plot in a surprisingly short amount of text. In short, I enjoyed it. Here are some things that you can tweak to make this story read smoother, as well as personal preferences that you might consider changing.
“Change the song; it's making me weep.” Weep is a strong word. While adjectives are a blessing upon us all, the word "cry" might better fix the context here.
"...time and place, and sometimes that place is the last place you want to go." Word repetitiveness is a little pet peeve of mine. The first two words "place" work because you're trying to describe the same thing. However, you can phrase the last part to not use the word "place" for a third time. A suggestion:
...time and place, and sometimes that place is somewhere you never want to go.
"It had been a long time ago and of course it had been my fault." The phrase here "of course" makes the protagonist sound more like he's complaining about his bad lot in life. It could be omitted if you wished.
"The couple we had been waiting for knocked at the apartment door at that moment and off we went." I don't really understand this sentence. Who is "the couple"? Why were they waiting for them?
"Her quick smiles had a kind of sad hesitation at times, but it passed quickly." First of all, the phrase "kind of" isn't quite necessary here. Secondly, you could put another sentence after this, mentioning how this may have troubled the protagonist slightly, but it never really bothered him enough to care.
"She was almost desperate, wild to biting and scratching." Maybe it could be better phrased as:
She seemed almost desperate for me, wild with her biting and scratching.
“You are a vampire like me now.” Possibly the most important statement in the entire story. However, it's slightly awkwardly stated. Here's my suggestion:
You're a vampire now. Like me.
You could also put the name of the protagonist here, since I can't find it anywhere else.
You're a vampire now, Joe. Like me.
"She had been right too, after thirty or so years we grew apart. It’s the hunting all the time I guess. Love is dangerous for us." There are several things you can consider changing here, I'll just put down what I what have done:
She had been right. After thirty years or so we slowly grew apart. It's the constant hunting, I guess. Love is just too dangerous for us.
"Not growing old together, having children all the things the mortals have in common, we can never have those things." Run-on sentence here.
Being unable to grow old, to have children, to have all the things mortals can have. We could never have them.
Now, some extremely tiny issues:
Please put a period at the end of your summary/preview.
There are two spaces between "that" and "pull" at the beginning of the story.
For the last line, there might be a comma after the "says". However I'm not too sure about that.
I hope you don't mind that I'm being so picky here. It's really a compliment, because if I stayed general then I wouldn't be able to criticize at all. I hope my feedback has been helpful to you, and best wishes on all of your future writing!