I want to talk to this person that works at my school about what is going on in my life. Only thing is, its not "technically" their job to be a counselor. What are the chances they would want to sit there and listen and help me? Like do you think I would be annoying them, or do you think they would send me away.. or do you think they would be interested in helping me even by just listening?

Are they a teacher or the janitor, and then it goes down to what type of person they are, and if they feel they are crossing a professional line.
Is there a reason you can't talk to the real school counselor

Even if this person cares for you and wants to talk to you it is in their best interest to draw a boundary with you and lead you to a counselor. There are many legal issues that could get in the way for this person. It's not that people do not want to help. They may be able to listen but should direct you to your counselor. (This coming from a counselor myself!)

Why not give it a shot?
First try talking to them about something not so personal and see what will their reaction be!

If they're not interested, then stop look for the counselor or a very good friend.
If that person is a teacher, then really I don't advice you... teachers usually hate it when their students talk to them about personal stuff because they would have this unfair advantage (my mother is a teacher ;))

But my final advice, go talk to your friends! ;)

Best of luck ;)