You only need to give antibiotics only if the cough is due to bacteria infection. Most coughs are due to viral infection and do not require even the antibiotics.
Keep the child warm

Most infants do not expectorate, they swallow their sputum. Additionally cough syrup may be sedative and of dangerous consequences if mum does not present early enough for treatment of pneumonia.

Kindly note that cough syrup is not recommended for infants especially below the age of 1 year. The most important thing to do is to treat the cause of the cough