Learn English – What does it mean by the phrase “not the story to tell”

It's a sarcastic response.

Taken literally the sentence would mean the speaker isn't already familiar with the situation and wants more information. However it is rarely (these days never) meant literally, and is used sarcastically to mean the opposite. I.e. That the speaker is already familiar with the situation. Its meaning is similar to the expression, "You don't say?", where the speaker is acting like they don't already know, when in fact they do.

The difference between these two expressions however is that, "Tell me about it", has taken on a sympathetic tone over the years, indicating shared misery, whereas, "You don't say?", is an insult where you don't actually care what the person has to say.

There's little in the words themselves to indicate that difference. That difference is simply the connotations those expressions have taken on over time. Tone of voice is pretty critical here, because the difference between a sympathetic statement and an insulting one is simply a different inflection while saying the exact same words.

Lit is the past tense of light. It means that the stairs did not have good lighting (or more specifically, the person who set up the lighting on the stairs did a bad job.)

The joke is a pun that comes from the fact that lit or lit up can also mean drunk.