Here's a good bit of advice from Charles Krauthammer to GOP leadership. You can't govern from one chamber of Congress, so why even try? Instead, make things difficult, put the other guys in a pickle, win little tactical victories rather than lose the grand strategic ones.
I like Charles' idea here--give the Dems a short term debt limit expansion with only one string--the Senate has to pass a budget (which it hasn't done in years). This will make Senators vote on actual spending levels and policy preferences, which will be important in the 2014 election when there are a few vulnerable D's up for re-election.
Fight the fights you can win.