How well does the antibiotic cefadroxil work against acne?

Acne. Cefadroxil is of no use in acne and should not be used for this purpose.
Not first choice. Antibiotics are not first choice medications for acne because long term low dose use of antibiotics can breed resistance (which means the antibiotics won't work for more serious infections when you really need them). If you have serious acne, you should certainly seek a dermatologist to make sure you get the best treatment recommendations for your skin.