Задача

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.