How do I handle negative feedback during and after a Product Hunt launch?

Depends on what it is. I would respond to that person and turn him into a contributor. Without know what they wrote here is how I would handle. Hey Xxxx, thanks for your comment regarding xxxxxx. While we put our hearts into making cxxxx the best we can, there maybe something we overlooked. We appreciate you bringing this up. Would you be open to a call so we can understand more about your thoughts so we can make sure no one else has this experience? This makes them feel heard and one step further, allows you to learn more

Embrace negative feedback as a chance to improve, Irsa. Address each point thoughtfully and show how you're using the input to make your product even better.

Thank them, ask them more questions if you can or need to. Note it as feedback in a feedback doc. You can act on it if it seems pressing but if it's more an opinion and what they think would make it better, ask others if they agree before making changes.