Puppy colds aka kennel cough can last 3 days to 3 weeks. Respiratory issues with young puppies are very common. Same as a common cold that one child gives to another at school. The smaller the breed/puppy, the weaker the immune system to fight the infection.
Coughing may be annoying but it is therapeutic. Coughing brings up the pus, mucus, and inflammatory cell products, which make our puppies sick. If the secretions of the lungs are allowed to dry up, the puppy will never be able to cough them up. For this reason fluids must be maintained to keep the puppy hydrated and keep the respiratory secretions wet. The gurglier (wet) the cough the better. Patience is always a helping hand with a cold. Breeders are always more successful treating respiratory issues because of the lack of activity the young puppy gets.
The infection needs to run its course. Rest, feeding and also antibiotics help the run. The most important thing is to feed feed feed the cold. Make the food as attractive as possible to encourage the baby to eat. Let the puppy have access to food & water 24/7. Second, limit the amount of activity/play time. Have the puppy sleep as much as possible. For some relief from the coughing you may give the puppy some Robitussin DM. This will make your puppy more comfortable at sleep time. 1ml/cc is fine (1 eyedropper). Remember, you don’t want to stop the coughing – you just want to have the puppy more comfortable so it can sleep. Third are the antibiotics as prescribed by your veterinarian.
In my opinion the key for puppies to overcome colds is food! High quality canned food mix with the dry top-dressed with honey works magic. In most cases puppies can even overcome colds without antibiotics. Especially if there hasn't been any loss of appetite.