Snape might endure the punishment in stoic silence. He is a complex character and often used to hiding his true emotions. So, he could just grit his teeth and bear it, all the while plotting his revenge or a way to get out of the situation.
He could also try to reason with Voldemort. Snape is intelligent and manipulative. He might attempt to use his words to convince Voldemort that the punishment is unjust or that there has been a mistake. He could bring up his past services to Voldemort or offer some new information to sway Voldemort's decision.
One possible reason could be that Snape fails to deliver crucial information. Voldemort is a very demanding and unforgiving master. If Snape doesn't give him accurate details about the Order of the Phoenix's plans, for example, Voldemort would surely be enraged and punish him.
However, as the story progresses, Harry may start to understand the reasons behind Snape's actions. He could become more resigned and accept the punishment as a part of his growth. He might use the experience to become more cautious in his actions and more respectful of the rules.
At first, Harry is likely to be angry. He might feel that Snape is being unfair and targeting him. Harry has a strong sense of justice, and when he believes he's being wrongly punished, he'll be resentful.
Snape might initially be shocked. He's not one to show much emotion openly, but internally he could be very disturbed. He might start investigating what happened to Hermione immediately, using his skills in potions and magic to try and find a cure or at least understand the cause.
Snape could initially be skeptical. Given their long - standing enmity, he might not believe that Harry's apology is sincere at first. He might respond with a cold glare or a sarcastic comment, like 'Potter, your apologies are as empty as your head.'
He could survive through his strong will and Occlumency skills. By blocking Voldemort from fully penetrating his mind, he can keep his most important secrets safe and hold on to his sanity, which is crucial for survival.
In some fanfictions, Snape might punish Harry by giving him extra detention tasks like scrubbing the cauldrons in the Potions classroom until they shine. This is a common form of punishment as it's within Snape's domain as the Potions master.
Snape might be initially in shock. He had trusted Lily deeply, so the betrayal would hit him hard. He could become very angry and bitter, perhaps vowing revenge against her. He might also close himself off emotionally, not wanting to trust anyone again.
He is often very suspicious. He might immediately think it's some kind of trick by the Marauders or their descendants. He'll likely try to figure out how to use it to his advantage against those he dislikes.
In some fanfictions, Snape might see this as an opportunity for his own growth. He might accept the apology gracefully, realizing that holding onto grudges isn't helping him. This could lead to a new relationship between them, where they start to understand and respect each other, or at least tolerate each other's presence.