I had the opportunity to stay for six days at Grigoris' apartment, which is very close to Monastiraki metro station (45 minutes from the airport). It's a stylish, cosy and comfortable place and I felt very much safe and at ease. You are at the top of a building (there are two elevators) so the view of Athens is quite impressive, especially since there's a wall of glass in the stretched living room and you can watch everything very easily from almost everywhere. It was just a little too cold to sit and eat outside, but that probably adds some extra flavour to the whole experience.
The apartment is smack in the middle of a very busy part of Athens, but the noise outside was very doable (I used some earplugs as well, just to be sure). You can walk up to the Acropolis in 15 minutes, there are supermarkets within a few minutes (and a Starbucks within a minute of leaving the building), and if you like Souvlaki and Gyros you can literally eat your heart out. My overall experience is very positive, not in the least because Grigoris responds really quickly. I totally and warmly recommend this place!