Someone looking at their calendar for appointment dates might use this phrase, meaning there are no appointments before Sunday.

"There's nothing before Sunday", could mean that there is no bread, milk etc available or that an event like a concert does not begin until Sunday.

I would say it depends on the context. I wouldn't use the phrase by itself unless I was checking some kind of schedule to see if there was room for the activity you were suggesting. Like if you asked someone if they had some free time to go out for lunch with you some time soon. Another example would be if you were setting up a business meeting or an appointment, although in my country Sunday would be an unlikely day to do business.