A predator like that probably won't starve in a month. Its a bit hard to say, you could compare with a crocodile (larger ones can fast for 2 - 3 years) but they are cold blooded which is a huge metabolic advantage here.
Lions can go 2 weeks without eating, and are far smaller which means there's less of them to waste away before they perish. A massive predator like a rex would having huge amounts of mass to lose before it died.
Chances are they probably only ate every 1 to 2 weeks by default. A month is not much of a stretch for it, so you likely fail.