In what way was Eisenhower's approach to government similar to the ideologies of the Republican presidents of the 1920s?

There were some similarities between the America of the two post-war periods, but there were significant differences between the administrative styles and personal actions of Eisenhower and the 1920s Republicans. The economies were similar in that, unlike most nations involved in the world wars, the American economy was helped by increased industrialization.  The fact that the wars were not fought in the US meant the industries of the country were not subjected to the damage of warfare.  The vast size and manpower of American industry was massively amplified by the wars, and technological advances also helped the booming postwar economies.  This led to increases in wages and credit, and Eisenhower (like the 1920s Republican presidents) believed in allowing the business world to largely run itself, although he had no intention of allowing the sort of out-of-control capitalism the previous administrations had.  He realized that was the major cause of the post-war boom of the '20s turning into the Great Depression.

After both wars there were also "Red Scares," the first caused mostly by the Russian Revolution.  The second was caused by the so-called "China Lobby" blaming the "loss" of China on the Democrats.  Between the fall of the Nationalist Chinese and the Korean War, right-wingers such as McCarthy used the alleged subversion of America by "Reds in the government" to foment discontent with the Truman administration and the 20 years of Democratic leadership.  Eisenhower's management style was low-key, and he tended to work through intermediaries, not doing or saying anything controversial in public himself.  He did, however, unlike the 1920s administrations, eventually make his personal displeasure with red-baiting known.  He was, behind the scenes, a major player in the downfall of McCarthy.

He seemed to take little notice of the running of the government, like Coolidge, but in fact he was a very active administrator, and a master at choosing those to whom he delegated tasks.  The one exception was the vice president the party gave him, Nixon, a man Eisenhower came to loathe.  By the time he was into his second term Eisenhower had become disgusted with the way the Republican party was run, although he was very much a conservative.  Like Hoover, he was a man of great ability and sympathy with the average American who found himself in an atmosphere of political chicanery he abhorred.

Another similarity between Hoover and Eisenhower was a desire to mend fences with Russia.  Hoover extended some tentative economic help to the USSR, and Eisenhower and Kruschev both worked hard (behind the rhetoric) on finding a way to end the Cold War before it ruined the economies of both countries.  The U-2 incident, unfortunately, put an end to that.