Praise be to Allah.
The relationship between brothers and sisters is one that should be based on affection and compassion, in which the younger honours the older and respects him, and the older takes care of the younger and shows compassion to him, as at-Tirmidhi narrated in his Sunan (1919) from Anas ibn Maalik (may Allah be pleased with him) that the Prophet (blessings and peace of Allah be upon him) said: “He is not one of us who does not show compassion to our little ones and respect to our elders.” Classed as saheeh by al-Albaani in Saheeh at-Tirmidhi.
Al-Munaawi (may Allah have mercy on him) said:
The one who is older should be given his due of honour and respect. End quote from Fayd al-Qadeer (5/388).
With regard to the obedience that is required of a woman, this is only towards her parents and her husband, subject to the guidelines and conditions that are well-known in Islamic teaching.
With regard to the brother, whether he is older or younger, he does not have the right to his sister’s obedience so long as she is still under the care of her father, and he is not her legal guardian according to Islamic law.
He only has the right of guardianship over her with regard to some of her affairs, such as in the case of marriage, and the rulings that come into effect if there is no one else who has a greater right to guardianship over her. This has to do with the idea of men being in charge of women that is mentioned in the question.
One of the duties of being in charge is that the brother must take care of his sister, and spend on her maintenance as required, if she does not have wealth from which she can spend. He must also pay the costs of any essential medical treatment that she may need, if he is able to do that, and he should accompany her to the hospital, or to any place that she needs to go and needs someone to accompany her.
The Prophet (blessings and peace of Allah be upon him) said: “Each of you is a guardian and each of you is responsible for those under his care.” Al-Bukhaari (893) and Muslim (1829).
Perhaps it is because women have a great need for someone to look after them than men, and because they may be weak and unable to look after themselves, the Prophet (blessings and peace of Allah be upon him) instructed men to take care of women, especially as it says in the report narrated by al-Bukhaari (3331) and Muslim (1468): “Take care of women.” There are many well-known hadiths which convey a similar meaning.
With regard to your obeying your brother in day-to-day matters, this comes under the heading of kind treatment and upholding ties of kinship, which Allah has enjoined. It is not possible for family relationships and households to be run along the lines of “obligations”, “rights”, “the law” and “judgement.” No, a decent family life cannot be established on such a basis; rather mutual kindness, compassion, affection and good treatment of one another should prevail. The more a person upholds ties of kinship and shows kind treatment, the greater will be his reward and his status before Allah.
And Allah knows best.