Jennesy -I think you'll be disappointed staying way out there' also you'll have trouble getting around. NOBODY speaks English on local busses or Trams, you'd have no notion of where you were heading. Here is a Map (click to enlarge) http://www.orangesmile.com/destinations/img/athens-map-big.jpg GIZI is At top, under the #9.
Your decision depends on what your budget is (50? 60? 80?E per night???), what your length of time in Athens is, and you want to do with your time. If you have less than 4 days, and you want to sightsee even the most major landmarks ... you won't have time to "get a feel for daily life" anyway -- you'll be too hassled trying to figure out the transit system!! As for time on transport -- You need to be at the Airport5 by 6:30 for an 8:30 flight, It takes 1 hour from Syntagma Square b y express bus to Get to the airport ... much much longer from where you would be,because the metros don't start til 5:30 and your district isn't even near the Blue Line. And you'd need to get to Syntagma to take the x95 express bus. YOu may well end up spending many more €€€ because you'll need to taxi a lot! As for getting to Pireaus, you can do it in 45 minutes if you're near the Green Line -- but you would not be near that either. Here's a Metro map to show -- http://www.ametro.gr/files/images/AttikoMetro_Athens_Map_eng_LG.jpg I lived in Manhattan 35 years, and subwayed to Brooklyn, Bronx, Queens etc ... but everyone speaks English! Whereas I've stayed in Athens at least 15 times in my 11 greece trips... and I still don't manage the local busses very well.
As for life in a neighborhood, you can find that MANY places in Central Athens, if you keep your eyes open. Go back to that Map, and find the Acropolis. Look just to the right where there's a #79 (Ag. Ekaterini church). Right beside it is my wonderful budget Gem, Hotel Phaedra, 21 rooms, family run (and a room with PRIVATE bath across hall can be under 70E in May-June-September). From a balcony I look down into the church's sunken courtyard and during every stay, I see a wedding, a christening party, a funeral. A 200-yard stroll over to the pedestrianized promenade south of the Acropolis, where at dusk, every fine day of the year, locals are pushing their baby carriages and buying their kids balloons. One Friday in june, I walked south from that Promenade on a narrow street, and heard music, and cheering. It was the last day of school.... I stood outside the schoolyard fence, saw kids singing in front of proud parents, getting prizes, a clown performing, firecrackers going off ... neighborhood families celebrating another school year. Daily life goes on almost all over Athens .... and if you're in a convenient area where you can walk everywhere, you'll have time to observe it.