Does he mean well and am I over-reacting? What would be the professional way to handle this?
Yes, most likely they mean well, otherwise they wouldn't have bothered to give you this feedback.
Getting feedback is golden, even more if it's constructive. You were given some suggestions to improve your overall soft-skills in a professional environment. You should take it as a chance to reflect upon that feedback and see where you have room for improvement (if there is nothing to reflect on, or the "feedback" had no merit, then waste no time reflecting and move on).
It's 1000x better that you got this feedback now, somehow informally (but with the chance to act on it now), rather than more officially in a performance review.
I'd add to my answer that, well-intended or not, constructive or not, this coworker should have told OP this in private, and it was not ok to do it publicly.
I suggest that you make your coworker aware of this, and politely tell them that in the future to come privately to you with "feedback" like this.
However, if this is not the first nor second time the coworker does things publicly, then that would be more in the "escalate to your manager" territory.