Respiratory infections in new puppies are common just like kids at school with"colds". Kennel cough is a term that has many different meanings depending on whose using it. What you describe indicates irritation I the back of the throat and especially if his attitude and appetite are good he will get well but sometimes it takes 4-6 weeks. Using guaifenesin expectorant at o.1 ml per pound body weight 3 times daily as well as moist air treatments can only help. As long as he feels well you can begin that and see what happens over 4-5 days before deciding on antibiotic use. Please let me know if any questions.