In this example, I'd actually use "One night."

First, there's nothing grammatically wrong with "On one of those nights"; in fact the standard idiomatic usage "One of those nights" does imply the "on." But there are two idiomatic usages to look at:

• "One night": This is a common way to give an imprecise time. It often provides a setting at the start of a narrative: "One fine day, Peter opened the gate and went out into the meadow." You might even be less specific and say "One time, he bumped into his dad," which is equivalent to "once." The phrase "one night" works together as an adverb.
• "One of those ___." This is also its own idiom, which can work in several ways:
  • Especially if used with the future tense, it can just mean "at some unspecified time in the future": "One of these days, Alice, I'm gonna send you to the moon!" ("One day" can also be used this way: "I'll be rich one day.") For some reason, there isn't a corresponding use meaning "at some unspecified time in the past." It isn't really idiomatic to say "One of those days, Peter went out into the meadow" (though it would be understood).
  • "X was one of those Ys" can also mean "this X is a typical member of the set Y." "I'm having a rough time. It's been one of those days." (Shorthand for one of those awful, dreadful days, which I don't need to elaborate on because you know what I'm talking about.) Cole Porter contrasts eloquent poetic expression of romantic thrills with the vague, glib dismissal that "It was just one of those things."
• So, in your example, "On one of those nights" would make perfect sense if it were actually referencing some nights that had been literally mentioned in the previous sentence, like: "He crept out of his room more nights than not, when everyone was asleep. On one of those nights..." But otherwise, "One night" is the best way of setting an unspecified night.
• By the way, note that if you do use "one of those," then the following word should be plural ("nights").

As a general rule, we use 'this' for something immediately in front of us or brought to our attention, and 'that' for something that we are highlighting from a distance.

Example: Imagine you're in a crowded room. If a friend introduced you to someone - brought them to you - they might say "this is John". But, if your friend pointed at someone on the other side of the room, they might say "that is John".

The same is true of 'these' and 'those'. Generally, 'these' is for things brought to you, 'those' are things that are highlighted to you.

Example: If your friend handed you some photographs, he might say "these are photos of my family". But, if he pointed at some photos hanging on the wall, he might say "those are photos of my family".

There are exceptions. For example, if a group of people were in the distance but were intentionally heading towards you, one might ask "who are these people?". Likewise, someone might preemptively say "look at this" before bringing something to you. So, it isn't so much the distance between you and the thing being referred to, but whether it will remain remote.