The interview was recorded in Toronto, Canada, in September on the sidelines of the Invictus Games, the athletic tournament created by Harry for wounded former soldiers.
Despite admitting concern over the future of the US, Obama, who did not mention his successor Donald Trump by name, said he felt a sense of "serenity" on leaving the White House.
"There was a sense that we had run a good race," he added.
On his new routine, Obama told the prince: "I wake up later, it's wonderful to be able to control your day."
The prince opened the discussion by saying: "This is the first interview you've said 'yes' to doing since you handed over the reins", to which Obama replied: "that's true".
Harry is due to marry his US actress girlfriend Megan Markle on May 19 at St George's Chapel in Windsor Castle near London.
According to British press reports, civil servants are concerned that a wedding invite from the couple to Obama could cause a further rift between Britain and Trump following a series of high-profile spats.
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