Time to crack open the ole Amanac…
In 1928, Time’s “Man of the Year” was Walter Chrysler, founder of his eponymous auto company. So he would have been on the list at the time. Ditto Henry Ford.
The most famous man in America, and maybe the world, was Charles Lindbergh. He’d be on the list.
Admiral Richard Byrd had just flown over the North Pole (or so they thought) and crossed the Atlantic by plane. He was incredibly famous and respected at the time plus he started his antarctic expedition in '28, so he’d be on the list.
Hoover had just been elected President, so he makes the list (number one?) So would Coolidge and Chief Justice William Howard Taft.
Al Capone was already ruling Chicago. But I don’t know if he was nationally notorious yet (the St. Valentine’s Day Massacre was in 1929). I’d put him on the list, though.
And around that same time a lot of interesting research was going on with communication technologies. A few different people were filing their television patents: Philo T. Farnsworth, The G.E. and RCA corporations. But most people would not have been aware of the importance of their accomplishments in late '28.
Steamboat Willie came out in '28 too. Again, I doubt anybody would have recognized Walt’s achievement at the time.
The Jazz Singer came out in '27. So talkies would have been getting pretty popular. I don’t know who the top star of '28 was (Jolson, maybe) but he/she would be on the list.
In sports…
'28 was an Olympic year. However, the only athlete I can think of from those games was Johnny Weissmuller and he wouldn’t become incredibly famous until he started making Tarzan movies a few years later.
Baseball was in its golden age. Mickey Cochrane was named MVP, but The Babe was by far the most popular player.
Notre Dame was in its hey-day. Knute Rockne was the most famous man in college sports. He’d make the list.
Also, interestingly, there was no Nobel Peace Prize handed out in 1928. Must have been a rough year.