We had an awesome time in a cool 6th floor apt. in one of Athen's coolest neighborhoods. After 5pm the area comes alive with restaurants and bars full of people spilling into the streets. Easy walk to Monistiraki square which is where you can kick off various tours, catch the metro, grab a cab, etc.
The view of the Acropolis is amazing. The wifi is high quality (there's no tv so I suggest bringing a laptop of tablet for streaming) and the apartment has everything you need for a short stay.
To be fair, the neighborhood might not be for everyone. But I saw few tourists ..which is what I like. It felt like we were just getting lost in the daily activities of the locals.