Cough syrup is not safe in such a young infant. It also takes time to become better. Follow up with your paediatrician

You can consult again and also cough syp can be changed. Cough takes 5-10 days to settle

Below 2 months (infact 6 months) rarely fall ill. They get cough and cold only if someone in the family is ill or the child is on bottle feeding. If someone in family is ill, then they should avoid close contact with the child. If mother is ill, then should wear mask while breastfeeding, should do handwashing with soap water and avoid contact wherever possible. If the child is on bottle feeding then stop it immediately and the child's situation will improve drastically.