What grammar pattern is used here and what does it mean?

Solution 1:

And no laughing matter to them as had the doing for him, as I had—my sister being took up with her little ones so much

There may well be a better 'translation' or gloss but this is the gist of the sentence:

And it was no laughing matter for those of us who had all the looking after him to do, like I had - what with my sister being so wrapped up in looking after the kids.

Here's a kind of step by step gloss:

And [it was] no laughing matter for those [/them] who had the doing [all his stuff] for him [to do], like I had [to] - with my sister being taken [/took] up with her children [little ones] so much.

The section them as had the doing for him is a type of relative clause. In modern standard English, we prefer not to have an accusative pronoun like them as an antecedent for a relative clause - hence those in the modern gloss above. The relative clause is fronted by as instead of a modern subordinator such as that or a relative pronoun such as who.