You are talking about the regular budget process, of which we have not had since 2009. We are under a Continuing Resolution right now, and Congress says they need another one in just a few days. That CR deal needs both the Senate and the House, and THEN it has to be signed by Obama. The GOP is now floating the idea of delaying ObamaCare, which would be a good thing. But the odds of the Democrats agreeing to that are horrible.
Regular budget process or continuing resolution, it takes both chambers to agree to continue funding spending at the current level. Lee, Cruz, Paul, Meadows, Gohmert, and others are prepared to fund all the government spending by the CR with the exception of ObummerCare.
The rub is that McConnell, Boehner, Cantor, et al. are too scared to fight. They want to roll over yet again.