Today I would bring you to visit 4 places of the ancient history in Athens.

I start with the Library of Hadrian, “overlooking” the square of Monastiraki. It’s located near the Roman Agora, and was built by Emperor Hadrian in 132 A.D. This library seems to be the widest, compared to the other Greek libraries existing in that period. It consisted of a large rectangular enclosure, which was accessed through an entrance located in the west of the complex. a garden and a swimming pool were included, by the emperor, in the area. During the sack of Athens by the Heruli, in 267, the library was destroyed. When rebuilt, it saw several changes of its intended use during the following centuries (during the Ottoman period, for example, it was the seat of the city governor). The ticket costs 4€, but if you bought (or plan to do so) the “cumulative” ticket which I have mentioned in the post about the useful information remember that the library is included in that ticket (I bought right at Ticket of Hadrian’s library).
Now I want to take you to visit the Ancient Agora: in ancient times this was the main square of Athens. Si trova a nord-ovest dell’Acropoli ed è delimitata a sud dal colle dell’Areopago e a ovest dalla collina di Colonos Agoraios. The ancient agora is located north-west of the Acropolis, south of the Areopagus hill and west from the Colonos Agoraios hill. There were private homes until it was reorganized in the sixth century BC, when Pisistratus made it the center of Athenian government. It was enlarged by Cimone, who built some new buildings. In the fifth century BC there were built the temples of Hephaestus, Zeus and Apollo. Very interesting is the Stoa of Attalo: the current building is a reconstruction/restoration of the old one and inside are exposed some remains from the agora.
Where, nowadays, are displayed these archaeological finds there were the agora shops. Within the area of the ancient agora there is also the Temple of Hephaestus: overshadowed by the Parthenon, it is one of the best preserved Doric temples among those that have survived to the present day.  It is also known as Theseion because in the Byzantine period it was erroneously believed to be the burial place of Theseus. The opening times of the ancient agora vary depending on the season: the summer is open from Monday to Sunday from 8:00 to 20:00 and in winter until 19.00. The tiket for this site (Ancient Agora + the museum), costs 8 €.  The next place where I want to take you is the Roman Agora
Located about 100 meters from the ancient agora, the Roman agora was built in Roman times, always by the Emperor Hadrian and was the forum of that time. Inside there is the Tower of the Winds: the inside, unfortunately, is under renovation and it isn’t therefore visitable  (there are no news about an expected reopening date, unfortunately). The Roman Agora is open every day from 8:00 to 15:00 (times posted at the entrance, at the beginning of May); the ticket costs 2 €, but it is also included in the combinated ticket (7 places for 30 €).
The last place I want to bring you to is the Olympieion, the Temple of Olympian Zeus.  Located about 700 meters from Syntagma Square, during the Hellenistic period and the Roman period it was the largest temple in Greece. Today there are standing only 15 of its original 104 corinthian columns and among these remaining 15 columns one is located on the ground: it fell when it was struck by lightning during a storm in 1852. Around the temple there are some remains of some ancient buildings. The Olympieion is open to visitors from Tuesday to Saturday, until 8 pm: I recommend you visit it in the evening, almost at the end of the day, to admire the sunset light. The ticket costs € 3 (but may vary and on the web there is no  official website). Just outside the area of the temple of Zeus, along the main road which goes from the center to the east of Athens, there is the Arch of Hadrian. It is thought that this arc, resembling a triumphal arch, was built to celebrate the arrival of Hadrian in Athens and to honor him for what he had done for the city (No, I haven’t become an expert in archeology and art history: I did some research, including guides, web and wikipedia before we left and even when I came back, to make sure not to write something stupid or false).
All these four places  may be reached and visited quickly and easily by foot, even in the same day. The only one whom may need more time is the ancient agora: it is the largest site and right for its greater extension needs definitely a little more time for the visit