At one point is used to describe the situation at one particular, unspecified, time. It is usually used close to the end of a scene in a story to provide additional information about what happened during that scene.
For example, imagine describing an afternoon outing, part of which involved a rowing boat and either incompetent oarsmen or drunk passengers. After telling the story of this scene you might add:
At one point, there were more people in the water than in the boat.
You are not saying when exactly in the scene this happened: it is completely unrelated to the sequence of events. You are simply providing additional information to help the listener or reader visualise what happened.
Note that it is not correct to start a story with
... a funny story happened to me.
The story is a description of what happened, not what actually happened. When telling this kind of story, the conventional word for the event itself is thing
... a funny thing happened to me.
If you want to start a story with a funny thing happened to me, you should really to give a general impression of when, and maybe where, it happened:
A funny thing happened to me at work yesterday...
A funny thing happened to me last week...
There is actually a musical and a film entitled A funny thing happened on the way to the forum.
Regarding right in your story: this is something that people sometimes say, but rarely write. It would work if this paragraph is dialogue- something that a character in your story says- but it would not look natural if it were a part of your own story.