Copernicus
Industrial Grade Linguist
The proverb is "Feed a fever, starve a cold", and uses an archaic meaning of "starve" which used to just mean "kill" or "die". It's a contraction of "Feed a fever and you will starve a cold", and originally meant that if a person has a fever, you should ensure that they eat an adequate amount, lest they succumb to a cold.I grew up hearing things like 'starve a fever and feed a cold'.
It's a single piece of advice, not a pair of alternatives.
It's been updated to include chicken soup.