TL;DR Camus's source of inspiration for Meursault's behaviour during and after his mother's funeral was very likely based on himself (or perhaps a close friend or relative; I'm still looking into that).
It is almost certainly a result of Meursault's being portrayed as an aphantasic character.
L'Étranger's story is written from a first-person point of view, so we know only what Meursault himself knows.
It is also written in an unusual French tense that makes it much more immediate and intimate.
The result is that we see what happens exactly as Meursault experiences it.
Had this been written in English, it would have been written in the present tense.
Most English translators don't understand this, and so tell the story badly.
The very first sentence is mistranslated, and as a result the whole meaning of the very first paragraph is lost.
And if one can't understand this first paragraph, one can't appreciate the rest of the novel.
The worst translations begin with "My mother died today", which is wrong in so many ways.
"My mother" makes it formal, cold, and impersonal; the original was the equivalent of "Mom", or even "Mommy".
Even those that don't make this mistake place his mother first, and then the rest of the paragraph reads as if it is about his mother and her death.
But it isn't.
In French, the book begins with "Aujourd'hui" (today), and that is what the first paragraph is about.
He is in effect saying:
Today is the day my Mom died.
The telegram said only that the funeral would be tomorrow, so perhaps it was actually yesterday.
He then goes on to describe having to ask his boss for two days leave and having to take a bus to the site.
The entire beginning of the story is about what he will remember about today, not about his mother.
Meursault's expressed concern is that his memory about today might be slightly wrong.
The fact of his mother's death itself is irrelevant.
Consider someone who:
- lives in the present, plans for the future, but ignores the past except when it becomes relevant.
- does not have memories that are burdened with sensual or emotional baggage.
- observes and analyzes every new situation, but remembers only the details that are important or different from the norm.
These are among the characteristics of aphantasia.
Meursault was almost certainly aphantasic (as was, I suspect, Camus himself).
Now consider some of the things that Meursault tells us about the bus ride:
- The trip lasted about 2 hours, and filled up along the route.
- He sat by himself in the same seat that he usually takes.
- The bus itself wasn't especially large or small.
- It made the usual number of stops to pick up or drop off passengers.
- The driver was neither female nor Japanese, but was experienced and competent.
- The road was in its normal state, neither recently paved nor in disrepair.
Only the "2 hours" is explicitly mentioned, and the filling up can be concluded by his finding someone sitting beside him when he awoke.
Everything else though, and much more, is implied by Meursault's not mentioning it.
What he reports is what he will remember, and what he remembers is only what he considers to be important or different.
That is how aphantasics think and live.
Is he upset that his mother died? Does he feel sad?
Of course, but that would be a natural response to the situation, so he doesn't see it as a significant fact worth recording.
Is he bothered that he doesn't know what day his mother died?
Yes, because when one has very sparse memories, not knowing something that might be significant is troublesome.
But it is only slightly so, almost nothing in comparison to the loss of his mother.
Aphantasics don't visualize past memories; they don't remember doing something, they remember that they did something.
Consider a boy falling down and breaking his arm.
He lies there feeling pain and fear, and seeing the look on his mother's face.
Years later, when he remembers the incident, he will re-experience that pain and fear and will re-see his mother's expression.
Had he been aphantasic, he would remember that he fell down, that his arm broke, that it hurt, that he was afraid, and that his mother looked terrified.
But, he will not feel the experience of the pain or emotions, and he won't even visualize what his mother's face looked like (then or at any other time).
Meursault's sadness and pain at learning of his mother's death was just as real and immediate for him as it would have been for any other person.
But Meursault doesn't recall the past by visualizing what he saw or felt; he doesn't re-experience anything that happened.
What he will recall is that his mother died, that he read about it in a telegram, that he went to her funeral by bus, that etc.
His mother's death is a fact that he has processed and remembered.
It is part of his past, and will be remembered only when needed, and only as if it were a third-person report.
When he is at the funeral, he still feels sad, but that goes without saying, so there is no need for him to comment about it.
What he does comment on is what is unusual or important, such as the annoyingly white walls, or the mourners that are there more to judge him than to mourn his mother.
Two days later, his memories of that trip have already become distant, accepted as reality and now simply a part of his history.
He will think about them, but when he does, it will be about the facts; he will not experience any emotional recall.
Of course he will continue with his normal activities (e.g. going to the cinema), because why wouldn't he?
The alternative would be sitting alone thinking about what just happened.
For most people that would mean experiencing hours of misery and hurt.
For Meursault though, it would mean two minutes of recalling every detail he remembers, without the attached emotions, and then what?
Why waste time feeling sorry for oneself when one has a life to live?
This is the cause of the seemingly uncaring behaviour (unusual from a phantasic's point of view) that was used in the trial.
For more details about aphantasia, see this personal description by Blake Ross (former director at Facebook and co-founder of Firefox): Aphantasia: How It Feels To Be Blind In Your Mind.