I'm no Bush fan, but the unemployment rate when Bush left office and Obama took office in January 2009 was 7.7%. (Your link shows 7.8%, but it was adjusted down to 7.7% the next year.)
You are right that Reagan did inherit a declining unemployment rate and a declining inflation rate. It was just too little too late for Carter, however, and he simply didn't have Reagan's appeal. And then, of course, the Iran hostage crisis was just a killer for Carter. But after he took office, Reagan's policies, particularly the tax cuts in 1981 seemed to cause a steep rise in the unemployment rate. The unemployment rate stayed high until, with the help of Dems in Congress, a tax cut was passed in late 1982. Finally in spring of 1983, the unemployment rate started to go down.
You are also trying to compare apples and oranges. You would have to compare the first 30 months of Bush, Reagan, and Obama to have a valid comparison. All three inherited a difficult economy; the dot.com bubble had just burst when Bush took over, but Obama definitely inherited the worst of the three situations.
So let's take the first 30 months of Bush, Reagan, and Obama:
First 30 months: Reagan: 7.5% to 10.1% Up 35%
Bush: 4.2% to 6.3%. Up 50%.
Obama: 7.7% to 9.2%. Up 20%
So Obama does beat both Reagan and Bush when looking at their first 30 months of the nemployment in their administrations.
Same as above, the data.bls.gov page. Also general knowledge.