Sorry to hear that your horse has developed a cough. A cough can develop from a number of causes. The most common of which are upper respiratory tract infections or inflammation. I would consider having your veterinarian out to evaluate and auscult your horses lungs, and even perform a rebreathing exam as we can have lower respiratory tract infections, or even chronic airway disease that can lead to a cough. In which case antibiotics or other medications (Clenbuterol - a bronchiole dilator) may be recommended.
Depending on the cause, the cough tends to be one of the last clinical signs to develop but also one of the last to go away. After a respiratory infection I have had patients whose cough takes multiple weeks to resolve even though they clinically appear normal (except for the cough). Most viral infections (upper or lower respiratory) tend to resolve over 3-5 or 5-7 days, but again the cough may persist longer.
A few things you can do as an owner to help are to keep your horse in a well ventilated area, and wet down any hay or pelleted diet they are fed. This helps decrease the dust particles and other allergens that can stimulate or exacerbate an already inflamed pharynx of your horse. I also recommend removing the horse from work as the more we make their lungs work, the more inflamed they can become, making the cough that much worse. Also monitor for any excessive nasal discharge, or color changes to the nasal discharge. All horses should have a small amount of clear nasal discharge from the nasolacrimal ducts which drain the tears from the eyes. Also monitor you horses temperature daily (normal = 98.5 - 100.8F) for fever. This can be a signs of a viral infection or a bacterial pneumonia developing.
There really isn't many medications that you can give to help reduce the cough, that may not test by next week. You can try using a short course of Phenylbutazone as an anti-inflammatory. The majority of other medications which we can use to treat a cough generally have longer withdrawal times (guifenacin, Clenbuterol, Albuterol, ect.). Your veterinarian may prescribe a short course of antibiotics to help prevent a secondary bacterial infection.
I hope this has answered your question fully. If it hasn't or if there is anything else I can answer for you, please do not hesitate to ask.