Start from the Greek parliament (you can easily get there with the Metro blue line that comes straight from the airport) and walk down Ermou street, a popular shopping pedestrian road. Once in Monastiraki, walk down Pandrosou street where all the tourist shops are and then walk up the ruins to reach an old Athens settlement with small houses. Then walk back down to Thisio and take Dionisiou Areopagitou pedestrian road to reach the Acropolis museum.
Syntagma Square is the central square of Athens. The square is named after the Constitution that Otto, the first King of Greece, was obliged to grant after a popular and military uprising on 3 September 1843. It is located in front of the 19th-century Old Royal Palace, housing the Greek Parliament since 1934. Syntagma Square is the most important square of modern Athens from both a historical and social point of view, at the heart of commercial activity and Greek politics. — Wikipedia
Monastiraki is a flea market neighborhood in the old town of Athens, Greece, and is one of the principal shopping districts in Athens. — Wikipedia
Anafiotika is a scenic tiny neighborhood of Athens, part of the old historical neighborhood called Plaka. It lies in northerneast side of the Acropolis hill. — Wikipedia
Thiseio or Thissio is a traditional neighbourhood in downtown Athens, Greece, northwest of the Acropolis, 1.5 km southwest of downtown. The name refers to the Temple of Hephaestus, which was mistakenly known as Thiseion, in reference to Theseus, the mythical king of Athens. — Wikipedia
The Acropolis Museum is an archaeological museum focused on the findings of the archaeological site of the Acropolis of Athens. The museum was built to house every artifact found on the rock and on the surrounding slopes, from the Greek Bronze Age to Roman and Byzantine Greece. It also lies over the ruins of part of Roman and early Byzantine Athens. — Wikipedia