As Salaamu AlaikumI have one elder sister who sometimes talk with me arrogantly and due yo this we fight with each other and I always apologize on the end due to which she is not changing her arrogant behaviour and I thought not to say sorry.Is this the right thing to do?

All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.

If your sister is treating you with arrogance that results in harming you as you mentioned, then there is no doubt that she is wrong by doing so; hence, we first advise you to supplicate Allah to reform her and make her regain her senses and rationality. Supplication is one of the best things by which a Muslim achieves what he aspires for; Allah Says (what means): {And your Lord says, "Call upon Me; I will respond to you." Indeed, those who disdain My worship will enter Hell [rendered] contemptible.} [Quran 40:60]

You should advice her in kindness and in a good manner, as the essence of religion is advise as proven in the Hadeeth narrated by Imam Muslim in his Saheeh.

If advice stems from compassion, kindness, and gentleness, it is more likely to be accepted. ‘Aa’ishah narrated that the Prophet said: “Whenever gentleness is found in something it beautifies it, and whenever it is withdrawn from something, it leaves it defective.” [Muslim]

If she takes head with the advice, then praise be to Allah; otherwise, it is permissible for you to abandon her if you hope that it will be better for her; otherwise, it is better not to abandon her.

Similarly, if your apology to her is beneficial, then you may apologize to her, and if it is not beneficial, then you may refrain from doing so.

Allah knows best.