Hello, my name is***** and I have over 20 years of experience as a veterinarian.
I'm sorry to hear that Prince has a cough and I understand that you are concerned about him.
A cough simply means he is experiencing airway irritation.
A cough can be related to heart disease, including heartworm, lung disease secondary to a viral, bacterial, fungal or parasitic infection, poor tracheal function, or tracheal irritation due to a bacterial, viral or fungal infection.
In a young dog infectious disease is by far the most common cause. Of those kennel cough or infectious tracheobronchitis is one of the most common causes. Because we know that he hasn't been exposed to other dogs and your other dog at home isn't coughing then kennel cough would be lower on my list of possible problems. It is still a possibility however as you could have carried the infectious agents home on your shoes or clothes, and older dogs with weakened immune systems can show symptoms whereas younger, healthier dogs may fight off a low level exposure.
If he has been vaccinated that's good but we cannot vaccinate against all of the agents that cause kennel cough and some dogs will still get sick even with the vaccine. The good news is that dogs that are vaccinated usually don't get as sick as dogs that haven't been vaccinated.
If this has been going on less than 2 weeks as long as he is still able to eat and drink well, isn't lethargic, isn't running a fever (rectal temperature greater than 103F), and he doesn't have a green or yellow nasal or eye discharge then you can try taking a watch and see approach and using an over the counter cough suppressant.
Make sure he gets plenty of fluids, lots of rest as the more active he is then more he will cough, and eats well. Add warm water or low salt chicken broth to his food. The better hydrated he is the better he will feel and the thinner his respiratory secretions will be.
You can take him into the bathroom with you when you take a hot bath or shower so he can breathe the steam which is soothing to his airways and thins mucous, making it easier for him to breathe. You can also use a humidifier in the room he sleeps in,
His cough should slowly resolve over the next 2 to 3 weeks if it is a viral infection. Most of the time these dogs feel pretty well so it's a matter of keeping them quiet and suppressing the cough if we can.
If the cough is very bad you can use human cough/cold medications with Guaifenesin (an expectorant) and Dextromethorphan (a cough suppressant) only such as Benylin or Robitussin DM can help. Give 1 teaspoon per 20 to 25 pounds every 6 to 8 hours. That would be 4-5 teaspoons or 1 &1/2 tablespoons for a pup his size every 6 to 8 hours.
If he is running a fever this is almost certainly a contagious infection and if he is running a fever and is lethargic then an examination is definitely warranted as pneumonia may have developed. Other infections must be considered such as fungal infections in hot, humid climates like Costa Rica. At his age other causes too, like heart disease or laryngeal paralysis, or even cancer, must be considered too.
Best of luck with your fellow, please let me know if you have any further questions.