"As I sit here ill, is there any credence to 'feeding a cold and starving a fever?' or is it the other way around?"First of all, get better, my friend. As for the advice, nope.
Medicine worked for centuries under the humour model (not to be confused with Comedist medical theory which focuses mainly on the sounds of digestion) in which health was based on a balance of moist and dry and warm and cold. Different bodily fluids had different properties with respect to these factors and if you had an imbalance of fluids, it caused physical and mental instability. We still have notions left over in contemporary language from this view -- we still call people sanguine or phlegmatic.
It was thought that if you had too much or too little of certain substances that it would create an excess of heat thereby causing a fever or a deficit of heat causing a cold -- which is why we call it that. Doctors would try to cure the ailment by bringing the amount of fluids into balance. This is why doctors used to bleed people with leeches, it was thought that an excess of blood caused certain ailments.
It was also believed that food created warmth within the body. So, if you have a cold, it means you are warmth deficient and food, in creating heat, would help to reestablish balance. If you had a fever, it meant an overabundance of heat and thereby eating and adding to the heat would only make the imbalance and therefore the illness worse.
Of course, we know that colds are actually caused by exposure to a rhinovirus and the effects are ways of trapping them in mucus and expelling them from the body and that fever are usually the result of infections and is the body's way of creating a hostile environment which makes them less likely to propagate and easier to kill. So, feed a cold starve a fever is not the way to go, but eat healthy foods as your body wants them and make sure to stay hydrated in either case.