Oldest spoken language in the world? All equally old.
Oldest still spoken language in a particular area? It's easier to answer this for islands or archipelagos, e.g. Great Britain: Welsh. New Zealand: Maori. Japan: Ainu (but maybe not for much longer).
Oldest written language? Sumerian (extinct) or Egyptian (not extinct, despite what the article states).
Oldest written language still in use? Coptic, a direct descendant of Egyptian used as a liturgical language in the Coptic Church, but not spoken as a vernacular for centuries.
Oldest written language still spoken? Hebrew (originally Canaanite), in continuous use but not as a vernacular for about 2000 years.
Oldest written language continuously spoken: Greek, but it wasn't continuously written. (There was a gap between Linear B and alphabetic script.)
Oldest continuously spoken and written language: Chinese.