How does Achebe vividly convey Obi's changing feelings towards Nigeria in No Longer at Ease?

The answer to your question has to do with Achebe's ability to use both direct and indirect characterization. Achebe uses these methods of characterization to highlight the character of Obi and how he changes throughout the course of No Longer at Ease. Achebe sometimes directly tells us about Obi’s changing feelings about Nigeria (which is part of direct characterization). Other times, Obi speaks and acts accordingly (which is part of indirect characterization). With direct characterization, we are told about the personality of the character. With indirect characterization, we are shown the personality of the character.
Obi is enamored with Nigeria at the beginning of the novel. Achebe uses both indirect and direct characterization to show this to the reader. Obi is happy to be in London studying to better his life and his education; however, he misses his homeland. The narrator directly tells us about Obi’s feelings:

Four years in England had filled Obi with a longing to be back in Umuofia. This feeling was sometimes so strong that he found himself feeling ashamed of studying English for his degree. He spoke Ibo whenever he had the least opportunity of doing so. Nothing gave him greater pleasure than to find another Ibo-speaking student in a London bus. But when he had to speak in English with a Nigerian student from another tribe he lowered his voice.

Through this direct characterization, the reader can see that Obi "longs" to be back in Nigeria. This example of direct characterization is especially pertinent in that Achebe actually uses the word “feeling” multiple times in regards to Obi. Obi even feels "ashamed" when he considers himself to be studying English instead of Law. Obi is so proud of his idealistic Nigerian ideals that he lowers his voice when speaking English to another Nigerian student. This is an example of direct characterization. Achebe tells us of Obi’s exact feelings here. Achebe describes all of this to the reader instead of using Obi's own words. However, there are some examples of indirect characterization when Obi recites his own poetry about Nigeria. These poems reveal Obi’s idealism about Nigeria through Obi’s own word choice. In these poems, Obi focuses on Nigeria's natural beauty and the honorable values among its people. At this point in the book, Obi truly wants to believe that the corruption among the learned in Nigeria can be overcome.
By the end of No Longer at Ease, Obi has lost his idealism completely. Obi has gone back to Nigeria and become one of those corrupt academics by accepting bribes. Obi has set aside his idealistic poetry and has just tried to "get by." Obi tries to convince himself that his financial situation excuses the "necessary" practice of accepting bribes. At one point, Obi tells the reader about his disillusionment and lack of idealism that he once had:

The impatient idealist says: "Give me a place to stand and I shall move the earth." But such a place does not exist. We all have to stand on the earth itself and go with her at her pace.

Obi, in fact, is only an idealist at the beginning of the novel. His poetry and his actions at university prove this. However, here we are shown Obi’s feelings only through his own words instead of being told directly that Obi has lost his idealism. However, with the help of Obi’s own words and indirect characterization, he tells us that this idealistic "place" actually "does not exist." Obi has given up and become a passive character who accepts the corruption as it comes. The rationalization here is that Obi attempts to "stand on the earth" by going "with her at her pace." In this case, the earth's "pace" now requires Nigerian academics to participate in the corrupt practice of bribery.