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.

Sugar and menthol suppress the cough reflex. However, the relief provided by suppressing cough can be detrimental to resolving infection. Sweet Taste and Menthol Increase Cough Reflex Thresholds, Wise, Breslin, and Dalton 2015

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, Dr. Mas Takashima, Baylor College of Medicine. See also healthline.com.

I use salt water. A bunch of salt added to sterile water. Nothing gentle about it. Give 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. Congestion relief. As a bonus it shortens the length of your sinus infection.

I follow with sucking on strong menthol eucalyptus drops for further relief. Although some research is against this, I think probably due to the sugar content of the drops.

Although effective against bacteria, it's not effective against COVID, although some other rinses/gargles are. Although I don't know if they are safe. See The effectiveness of various gargle formulations and salt water against SARS-CoV-2, ationg et al 2021

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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.