Short answer
to withdraw from a situation or to quit participating in a project or task, etc., and to thereby cause a negative consequence for someone.
Longer answer
We often use verbs (including phrasal verbs which this could be considered) followed by "on" to add a meaning something like "blamefully causing a negative consequence for someone or something."
For example, "She went crazy on me." Would generally mean that I was depending on her company or assistance with something, so her going crazy caused me some problem or distress.
To pull out means to depart (especially by vehicle, which I assume comes from carriages or coaches being pulled by horses, for example) or to withdraw:
pull out
- To leave or depart: The train pulls out at noon.
- To withdraw, as from a situation or commitment: After the crash, many Wall Street investors pulled out.
Since the first definition means a vehicle leaving, it would not normally be followed by on unless it meant on time or on a surface: The car pulled out on/onto the dusty road.
We can guess that it's more likely you ask about pull out on with respect to definition 2, because the three words seem to go together and in a way such that the meaning would be less obvious to many English learners.
So, to pull out on someone in this sense likely means to withdraw from a situation or to quit a project or task, etc., and to thereby cause a negative consequence for someone.
Examples:
She promised to invest $2 million in our company. We spent weeks preparing plans and drafting legal agreements in preparation for that. Then she pulled out on us at the last minute. We were so disappointed.
They were engaged for three years. Then John pulled out on her on the wedding day. He just left town without telling anyone.
He promised to help me move all of my things to my new apartment, but then he called and pulled out on me, saying he was too tired from work.
There are several questions packed in here:
1) How much notice would a moment have? There is some reason that the phrase evolved to "moment's notice" rather than "a moment OF notice", but I was unable to find the history of the phrase to further clarify. Looking at it practically, I would not think that a moment would have very much notice. Further, since being poetic is giving words life beyond their obvious meaning, this is definitely poetic.
2) How did something poetic become common usage? Given that the use of the phrase seems to really catch on after 1791, I would assume that something that was popular used the phrase. Some gentle searching of famous works at that time (particularly poetry) did not reveal use of the phrase, so at this point anything I say would just be speculation.
3)a) (implied question) Is the possessive correct? It is but not for the reason you think. My research suggests that 24 hours' notice is incorrect, but moment's notice is correct. As you can see in the graph, "24 hours notice" is the older phrase and given that they both persist (despite hours' being currently and generally more popular), I usually go with the oldest being the winner. Further, the interpretation that "24 hours" is a measure of the notice seems reasonable. This also, I think, proves that this question is not a duplicate given that the phrases are different. The difference between the two classes of phrase is clear given the usage pattern moment's notice versus moments notice. Finally, it's clear that possessives are extremely complicated, so I would think that explains the lack of definitive discussion.
4) (Implied question) What about the beginning of the phrase? If you accept the usage data, the correct phrase is "at a moment's notice", at least according to the ngram viewer. I don't have a great answer for why though.
5) (Implied question) Is the moment personified? Yes. Unless you only use a narrow definition of personified. If you don't consider the relationship between the moment and the notice, the phrase makes no sense. Therefore, we must consider how much notice a moment would give, and to do that, you have to think about a moment having attributes like patience and forethought, which is very person-y. Ironically, I think the problem came when @FumbleFingers "misparsed" @RocketBouchard's sentence "The moment is personified and it is noticing." I think that the OP meant to use "noticing" as a verb as in "providing notice". Probably not the best choice when asking this kind of question. It also should be clarified that this personification is only happening with specific phrases like "moment's notice". Car's window, for instance, doesn't require us to know (or imagine) anything about the car or how it relates to the window.
Best Answer
You have 90% answered your own question. 'So' is used as an adverb to mean 'so much', 'extremely', 'to such a degree', and dictionaries specifically mention 'so' used at the end of a sentence to mean 'to a very great degree'.
So, you can use it with a variety of nouns.
All of which said, in my experience it is a phrase which is, as you mentioned, more common in older songs and literature than in modern speech. To my mind it gives the impression of an Enid Blyton book from the 1930s.
Today I would expect the average youth to say: