Based on the personality traits you've described, it seems likely that he would have a complex relationship with his crush. His confident and confrontational nature might lead him to stare more than he realizes, as he could be both curious and intrigued by her. However, his rude and sarcastic tendencies could mean he might try to play it cool or act disinterested, especially if he's trying to maintain a "tough" exterior.
His nosy and opinionated side might prompt him to observe her closely to learn more about her, but he may also feel a bit moody or conflicted about his feelings, which could impact how often he actually stares. Overall, it's possible he stares frequently, but mixed with moments of acting aloof or teasing her if they interact, showcasing his more sarcastic side.