A lovely area, and very safe (BTW, the only area in Central Athens that isn't so safe is Omonia area -- others like Gazi or Psirri might be a bit grungy or rowdy at night but not "unsafe") ... Koukaki yes IS a longer walk to the Acropolis So. side entry gate... See this super online map http://www.orangesmile.com/destinations/img/athens-map-big.jpg (click & it gets huuuge; navigate with sliders along edges). Look at your google map -- maybe it tells how far from Acropolis, I cn't say because no address... anywhere from .5 miles to 1.5 miles I'd guess. After Acropolis, many people descend to Ancient Agora, then walk thru Monastiraki Square(don't waste time on Flea Market its just trashy souvenirs now)... have lunch, then make a discovery (like Melina Mercouri Foundation... a thrill! for women!!) -- you might want to taxi back to home base & put your feet up!
It's a low-traffic area, enjoyable walking, definitely family residences rather than 1 hotel after another. You'll love the pedestrianized promenade S. of Acropolis -- before dinner, Greek families stroll up and down (In Italy called the "volta"), children on trikes, balloon sellers -- like a movie set. In the past, because lo-key residential, not deluxe, fewer restaurants or shops ... but guess what?? I just googled "Koukaki Neighborhood" and guess what? Now it's the new cool (hot) place!! http://www.blog.visitgreece.gr/koukaki-is-athens-new-hot-neighbourhood/ -- a whole clutch of cafes, wine bars & a great place for "meze" (that's greek for Tapas), plus shops for jewelry & funky clothes. You may have stumbled onto a trendy area!!