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.)