Saying Okonkwo is arrogant is quite a harsh judgement. He definitely had a huge chip on his shoulder, but that was because his father, Unoka, had been such a failure:
In his day he was lazy and improvident and was quite incapable of thinking about tomorrow.
Okonkwo was embarrassed by his father's poor legacy and was determined to be better than he. His desire was to be the complete opposite of Unoka. He lived in a society which valued hard work and strength and Okonkwo set out to prove himself such a man - one who was powerful and successful. His life was dominated by a fear of failure and weakness.
Okonkwo, in a sense, wished to restore the family's good name and, as such, recreate its legacy. In this undertaking, he became absolutely committed and that made him a harsh taskmaster. He expectedthe same kind of diligence and loyalty to his ideal from the members of his family and responded, mostly unfairly, if they failed to meet his near-perfect demands. It is this obsession which made him seem arrogant.
He achieved great success and grew up to be much admired and respected, far and wide. He worked hard to take care of his family and proved himself to be a more-than-capable warrior in battle and in the wrestling arena, becoming champion at age eighteen when he threw Amalinze the Cat, who had been unbeaten for seven years. This feat brought great honour to his village.
Okonkwo was a proud man, but not boastful. The text does not refer to any incident where he brags about his prowess, either as a wrestler, soldier, farmer or as the head of his household. He did however, have "no patience with unsuccessful men" and this may make him seem arrogant. It is just that these so-called men envied and resented him and to them he would obviously have seemed arrogant.
It is this pride, and not arrogance, which essentially lead to Okonkwo's downfall. He became very controlling and this lead to an incident in which he beat one of his wives for a trivial misdemeanour during The Week of Peace. In this instance it was not arrogance, but anger, which made him forget about respecting cultural tradition. He was severely reprimanded for his disrespect and was accordingly punished and had to atone for his crime by providing one she-goat, a hen, a length of cloth and one hundred cowries.
Okonkwo usually apologised when it was decided by the Elders that he had been offensive in some way or another and he readily paid for his indiscretions without any protest - surely not the sign of an arrogant man. So it was that his gun misfired during a funeral and the young son of the deceased was killed by a piece of shrapnel from the exploding firearm which pierced his heart. Okonkwo was banished to his mother's village, Mbanta, for seven years for having committed a 'female' crime. He willingly accepted this harsh punishment and rebuilt his life, eventually returning to Umuofia.
It was Okonkwo's pride and not arrogance, which led him to kill Ikemefuna, the boy who he loved as his own son, when it was eventually decided that he should be executed in recompense for the murder of a female member of their tribe. Okonkwo was afraid of being seen as weak.
Also, Okonkwo was one of very few tribesmen who had the courage to vehemently oppose the arrival and progressive dominance exercised by the missionaries. He was prepared to oppose them and killed one of their messengers when they came to impose their will by stopping a public meeting. He had the foresight to know that the arrival of the missionaries who imposed their religion and their government on them would eventually lead to the destruction of Ibo society. This was not borne out of arrogance.
Finally, Okonkwo committed suicide for he was not prepared to endure the punishment which would have been imposed upon him by those who had usurped his country, his culture, his traditions and his son. He would rather commit an abhorrent act within his own tradition, than be shamed by those whom he did not recognise or acknowledge.