Can 持续 have these 2 meanings?
Yes. It seems you understand the first sentence pretty well.

Regarding the second sentence, I think you're confusing 持续 and 足以. 足以 is the word that means "enough". 

这足以持续到周末 = This is enough to last until the weekend.

The key here is that 持续 indicates a persistence, something that continues on.