what can or should they do about Obamacare? The current unpopularity of the program is the reason it looks at least possible that they will not only hold the House but get a majority in the Senate. If so, what are their options?
The obvious one is to replace Obamacare with something that moves medical insurance in the opposite direction, towards something more like a free market. One elements of such a bill would be legalizing interstate sales of medical insurance, another equalizing the tax status of individual plans and employer provided plans, which probably means making expenditure on individual plans deductible. Sponsors could plausibly argue that the savings from abolishing the ACA will more than make up for the lost revenue.
Obama can and presumably will veto any attempt to repeal his pet program. The Republicans will not have enough votes to override a veto, so their only hope would be to get enough Democratic senators and representatives to go along. That is going to be hard, but not necessarily impossible, depending on just how badly Obamacare is doing. One critical actor will be Hilary Clinton. On ideological grounds she should be even more adamantly in favor of preserving the program than Obama—but on political grounds she may be looking for a way of avoiding the political fallout from its failure. Organizing Democratic support for something she can plausibly represent as a compromise might be one way of doing so.
There is another alternative. Suppose the House and Senate pass a spending bill with nothing allocated to continued implementation of the ACA. Obama can veto it and force another government shut down. But it is going to be much harder for him to blame a shut down on the Republicans if both houses have passed a budget and his refusal to sign it is the only remaining obstacle.