The House passed a budget already. I would bet that budget has a deficit. If it has a deficit, then that would imply that the Sec of Treasury has authority to borrow the money to cover that deficit, debt limit or no.
If the Ryan budget which passed the House passed the Senate and was signed by Obama(neither of which will happen) it would go over the debt limit, too and require an increase in the debt limit. There is going to be an increase in the debt limit we just don’t know the terms of it yet. Both Boehner and McConnell have publicly said so.