What does it mean when the right lower part of your back hurts after drinking alcohol and you're on an antibiotic?

Likely unrelated . Depending on which antibiotic you are on and the reason you are taking them and how much you drank the pain you are experiencing is probably unrelated. However the area in your back that hurts is near your kidney and if you drank a lot and the antibiotic is metabolized/excreted through the kidney then you may be overworking them. Or it may be muscle strain. Best not to drink when taking any meds.