When the proclamation reached the region of Haizhou, there was a man, surnamed Chen and named E, his courtesy-name Guangrui,

The notice reaches Haizhou 海州, in what is now Jiangsu, and finds our man. He is Chen E 陳萼 by given name and Guangrui 光蕊 by 表字, the courtesy-name a man took in adulthood and went by socially. That is why the novel calls him Guangrui almost throughout, and keeps 陳萼 for formal moments — such as the last page.

who, having seen the notice, went home at once and said to his mother, Madam Zhang:

張氏 is Madam Zhang, his widowed mother, named the way women usually are in these texts: by the surname she was born with, and nothing else. She is about to become one of the most important people in the chapter, and she will never acquire a personal name.

'The court has sent down its yellow proclamation, opening the Southern Department to select men of worth. Your son wishes to go and sit the examination.'

南省, the Southern Department, is the Department of State Affairs 尚書省 at the capital, so opening it means the metropolitan examination is on. 省 reads shěng here, and 應試 keeps 應 as yìng.

'Should I win even a minor post, I would honor my parents and make my name known, gain rank for my wife and standing for my sons, and bring glory to our gate — such is your son's ambition.'

Four four-character phrases in a row, and the first two are still current Chinese. 顯親揚名, make your parents visible and your name heard, comes out of the 《孝經》, the Classic of Filial Piety, where a son's fame is something he owes his parents rather than something he keeps. 封妻蔭子 is the same bargain in hard currency: an official's rank bought his wife a title and shaded his sons with hereditary privilege — 蔭, yìn, is literally shade, and 子 here carries its full weight as zǐ. 倘得, should he obtain, takes 得 as dé.

'This I report to you, Mother, before I set out.'

稟告 is reporting upward. You do not tell a mother a thing like this; you inform her, respectfully.

Madam Zhang said, 'My son is a man of learning. As the saying has it, "In youth one studies; grown, one acts upon it" — so indeed it ought to be.'

幼而學，壯而行 — learn young, act grown — is the proverb a mother reaches for when she means yes. 行 here is xíng, to put a thing into practice.

'But in going up to the examination, take care on the road; and once you have your office, come home early.'

赴舉 is to go up for the examination. Then comes the line that will ache later: take care on the road, and once you have the post, come home early. 得了官 takes 得 as dé, to obtain. He does obtain the post. He does not come home.