When Rep. Paul Ryan delivers the Republican response to President Barack Obama's State of the Union address Tuesday, many viewers will get their first look at a man whom GOP leaders are trusting to manage a central policy issue—how to cut the federal budget—that could shape the party's image for years.

While unknown to most Americans, Mr. Ryan, 40 years old, has established himself as a leading conservative thinker on federal spending, shaped in part by his early work for supply-side icon Jack Kemp.

Now, Republicans not only have made Mr. Ryan chairman of the House Budget Committee, but on Tuesday the House is expected to vote to give him unprecedented powers to force spending cuts for the current fiscal year. That authority will allow Mr. Ryan to act unilaterally in setting an overall spending level for the rest of the year, a job usually handled by his full panel.

Hours later, Mr. Ryan will speak to the nation in a televised address following Mr. Obama's remarks to a joint session of Congress. He was chosen for the role by House Speaker John Boehner and Senate GOP Leader Mitch McConnell.

In elevating Mr. Ryan, Republican leaders are taking what Democrats believe is a political risk. He has written an anti-deficit plan that includes politically explosive ideas—replacing Medicare with vouchers and allowing some workers to invest Social Security taxes in private accounts—that go beyond what even many Republicans are prepared to embrace.