Some supplements such as garlic, may help prevent or shorten viral infections. However, every viral strain is unique, so what may work for one strain may not work with another.

Other supplements to look into:
Astragalus Membranaceus1
https://examine.com/supplements/andrographis-paniculata
Zinc and elderberry lozenges or syrup
https://examine.com/supplements/pelargonium-sidoides, except that the “vast majority” of studies were funded by a company that sells a patented extract.

Theobromine in cocoa outperforms pharmaceutical based cough suppressants. Pharmaceutical cough suppressants are not much more effective than placebo, and I have no idea why people keep taking them, despite the lack of evidence for efficacy.

Saline Nasal Rinse

A saline nasal rinse gives your sinus a REAL reason to start dripping. Let it sting baby! By the time you've finished rinsing the inside of your sinus with salt water, your nose will be too exhausted with mucus production, and will give you a bit of a breather from having to blow your nose constantly.

It will also shorten the length of your sinus infection.

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It may not be good to fast during a common cold, because of studies showing that cellular autophagy metabolism promotes viral reproduction:
Autophagy induction regulates influenza virus replication in a time-dependent manner
Autophagy Promotes Replication of Influenza A Virus In Vitro
However, different viruses may exploit different cellular machinery, so this conclusion may not apply to all viruses.