So here is a riddle: how is it possible for someone born in the 21st century to be older than someone born in the 20th century?

Chronologicaly it isn't possible for living people.

Obviously, the person born in the 20th century could have died as a young child (stopping their aging as a living being) while the person born in the 21st century could be older.

Biologically, if the person born in the 20th century had spent a significant portion of their life travelling at relativistic speeds approaching that of light, they would have aged more slowly than the person born in the 21st.

In terms of number of birthdays, the 20th century person could have been born on 29th February, and would have had a quarter of the birthdays they would otherwise have had, although this clearly doesn't alter their actual age.

Ethnoculturally, the two people may observe different calendars from each other meaning that the 20th century by one person's reckoning could occur after the 21st century in the other's.

20th century is from 1901-2000 whereas 21st century is from 2001-2100. There are several possibilities, such as that of AlterEgo's answer and the whole Leap Year thing.

The guy in the 20th century died very young.

The 20th century person died at a young age.

You bought a house from Century 21?

OK,for just 1/16 of a second it made cents,dint it?

No?OK

Aw, you're going to do that leap year, Y2K, Mayra calendar UFO behind the comet thingy again, aren't y'all?

They are a born again Christian,