Kyle,
Excluding lakes-only bulkers, small local passenger ships and harbour craft, the oldest ship IN CLASS (and therefore assumed to be in service) is the sail cruise ship "Sea Cloud", built 1931. The pipelayer "Henry P. Lading", built in 1930 as the tanker "Esso Kobenhaven", might also be considered though she is not fully self-propelled. Several former large tankers built in the early 70s are still in service as FSOs/FPSOs though, as permanently converted, might no longer be considered 'ocean going'. As far as cargo ships are concerned, a number of 1st generation containerships built 1970 / 1971 are still in service (pariticularly in the U.S.), as are the LNG carriers "SCF Polar and "SCF Arctic" built 1969. Numerous Chinese-flag bulk carriers and general cargo ships built in the 1960s are also claimed to be in service but came out of class years ago. In reality, most are probably scrapped though it's feasible some remain in service on Chinese coastal trades.
Hope this helps.
Brgds
Phil