If you are suffering from a typical cold virus infection, then you can expect the symptoms to last for up to 1 week or so, at which point they should be getting better. NyQuil is one of several over the counter medicines that contain sedating anti histamine and cough suppressants that are used to treat the common cold. It is usually taken at night to help with sleep while the cold and cough symptoms are at their worst. It is fine to take it for this reason.
However, if by 1 week the symptoms are not improving, especially if you have fever
or if the cough is getting worse or if you have any shortness of breath or chest pain, these could all be symptoms that the cold has worsened and may have evolved into a bacterial infection. If you have these symptoms, it would be worthwhile to go see your primary care doctor
for a quick checkup.
They will be able to determine if you have just a cold, or if you have something else. If your symptoms are from an infection requiring antibiotics or other medications, they will be able to prescribe those for you.
Good luck, and I hope you start feeling better soon!