Next time you should take the metro! It's clean, modern and efficient and easy to use and cheaper than taxis. Generally very safe but of course just be careful, as you would be in any other big city using public transport.
Also although the Plaka/Monastiraki areas are quite touristy, a lot of local Greeks also hang out there and go to the restaurants and rooftop bars, so it's probably more authentic than you think, but I get what you mean.
I've never been to Olympia and I used to want to go, but to be honest most people say it's not that impressive . If you go to Peloponnese (area where Olympia/Patras are), it's better to go to Nafplio (our former capital city) and the Mani Peninsula as well as the western Messinia area (Voidokilia Beach, Polylimnio Waterfalls).
Yeah I get what you mean about football and basketball, though it depends on which teams are playing. Games with other European teams are usually more calm (with some exceptions especially if the opponents are from the Balkans). I would avoid a derby at all costs...