The House does have the power of the purse (they are the only constitutionally authorized originating entity for spending bills, and it should rightly defund anything and everything with respect to Obamacare.
The only thing I worry about appropriations-wise, if a budget bill is passed, it will have to go through a House-Senate Conference which reconciles Senate appropriations with the controlling House bill and then a final House vote on whether to accept or reject the report.
This is where I feel we will get shived in the back by our own RINO House conferees, and the level of what they finally agree on will be buried in gobbley gook wording buried in the conferee report. Most House members will never read the conferee report and cross reference back to both bills.