How long does it take for antibiotics to kick in for a sinus infection?

Depends. Most "sinus infections" are actually caused by viral symptoms so wouldn't be helped by abx to begin with. If you're not feeling better after 2-3 days on abx, talk with your doctor and ask if this could be a viral illness. Also, most recent recommendations say that the zpack shouldn't be used first line for true sinus infections anymore.