Should I agree to marry my paternal cousin, who is ten years older than me, and is religious and well-mannered, but arrogant, bearing in mind that I hesitate to accept him when I see him, but I tend to agree to him when he is out of my sight?

Self-reliance is a virtue, as long as it does not reach the level of arrogance, conceit, and belittling others. At that point, it becomes a reprehensible fault, and its possessor needs someone to advise him on humility and clarify the harms of this trait. The sister can accept the young man if he is a person of good religion and character, while attempting to advise him gently and explain the danger of this trait, or she can reject him until he abandons it. As for her wavering opinion of him, we advise her to perform Istikhara (prayer for guidance) and supplication.