Ah, Anon, Anon…
A newspaper by any modern understanding of the concept presupposes widespread literacy, and, you know, paper. The Roman Acta Diurna were a daily gazette of government decisions published, Asterix style, in stone, and there may even have been equivalents in Greece for publishing what the assemblies had decided that day; but they really aren’t quite what you’re after.
Newspapers hit the Greek-speaking world at the end of the 18th century. Wikipedia credits Efimeris (1790) as the first; Hermes o Logios was certainly the first influential one. And newspapers flourished during the Greek War of Independence (the first newspaper in Greece started August 1, 1821 in Kalamata: Σάλπιγξ Ελληνική). Those papers were not in the vernacular—nothing was; but neither were they in Ancient Greek: they were in Katharevousa, because someone other than classicists had to be able to read them. And I’m reasonably sure that’s been the case throughout the 19th century, and up until the 60s. (What was the last Katharevousa holdout, Estia, is now the last polytonic holdout.)
The most excellent and commendable Akropolis World News is probably what you’re after: it’s a weekly post of a paragraph of world news, in actual Ancient Greek.