The passage you quote is taken out of context. But in this whole chapter we see Mrs Bennet and her daughters wishing that they could be introduced to their new neighbour, Mr Bingley. Mr Bennet, unknown to them, has already been to introduce himself to Mr Bingley, but he's pretending to his wife and daughters that he hasn't (as he very much enjoys teasing his wife). Your passage is immediately preceded by this exchange:
"When is your next ball to be, Lizzy?" "To-morrow fortnight." "Aye, so
it is," cried her mother, "and Mrs. Long does not come back till the
day before; so it will be impossible for her to introduce him, for she
will not know him herself." "Then, my dear, you may have the advantage
of your friend, and introduce Mr. Bingley to her." "Impossible, Mr.
Bennet, impossible, when I am not acquainted with him myself; how can
you be so teazing?"
From this, you can see that Mrs Bennet is desperately trying to think of ways to wangle an introduction to this eligible young man (ie via Mrs Long and her family), but Mrs Long is away and therefore won't be able to do the introductions.
Mr Bennet then begins to reveal the truth (ie that he's already made the acquaintance of the new neighbour), by saying that his wife will be able to introduce Mr Bingley to Mrs Long. Mrs Bennet doesn't understand how that's possible, as she hasn't met Mr Bingley herself.
Mr Bennet knows very well what she means and that she's confused, but he pretends that she is being very discerning and making a clever comment about how she cannot possibly know Mr Bingley well, as she will only have known him for two weeks by the time of the ball.
SO. To return to your question.
The 'office' that Mr Bennet is talking about is the task of introducing Mr Bingley to Mrs Long at the ball that will take place in two weeks' time. He says that HE will introduce Mrs Long to their new neighbour. He will 'take on' the responsibility himself. He does this because if he doesn't, someone else will: the absent Mrs Long and her (presumably single) daughters must also be given the chance to meet their new eligible bachelor neighbour.
(See the extracts from Pride and Prejudice.)
They argued a little about Sophocles, then in low water.
(Source, suggesting that the OP's transcription incorrectly ommited that final period.)
It means that in those days their school's curriculum allocated less time to the plays of Sophocles than was customary at other times in the school's history. The implied metaphor is that curriculums follow fashions, which rise and ebb like the tides at a beach.
Best Answer
Because of certain humorous cultural stereotypes associated with the ancient Greeks and Very Young Men, it might be best that I dispel any inchoate notions forming in your head.
The Very Young Man muses on all the wonderful opportunities such a Time Machine could afford him. He could learn Classical Greek from Classic Greeks!
However, if he did that, The Time Traveller replies, his teachers would plough him for the Little-go.
In the UK, plough was once used as slang for "fail":
And the Little-go was an informal term for "Responsions"¹, one of the three tests all students of the University of Oxford once needed to pass to earn a degree. Per Wikipedia:
Of course, knowledge of Ancient Greek being a staple of a classic education at the time, the Little-go would test The Very Young Man's knowledge of it.
And, just as obviously, if he had learned Ancient Greek from actual Ancient Greeks, directly, he would have failed that test.
Because, of course, modern teachers don't know how actual Ancient Greek was spoken; they only know the reconstructed and speculative version the "German scholars have improved upon".
And if you don't believe me, maybe you'll believe The Schmoop:
Now get your mind out of the gutter.
¹ Speculation: Little-go because the test was a go (attempt) at your degree: the first, easiest, littlest of the three.