hey mate,
I can't really help you on the places in Athens and Cyclades because I live in northern Greece. But I can give some info about the matches you want to watch.
About EURO Tournament, matches are at 19:00 and 21:45 local time and its on the national channel (ET1) so you will find plenty of cafe or bars to watch the matches along with a beer :P
About the NBA Finals, because of the time difference, the matches usually are at 02:00 after midnight (or even later), which means its highly unlikely to find any open bars at that time. Maybe any sports cafe, but I highly doubt that also... BUT the matches are on Novasports channel (usually on Novasports 2 or 3). This is a greek satelite channel, it is not a free channel, but the majority of hotels usually have them and offer them free of charge on their customers (but you could ask that in case you really want to watch them).
P.S.: Just one hint about the rooms you talked about. When you will arrive in the islands of Cyclades you will see dozens of signs saying "rooms to let" or even people coming at you offering you a room. These are the rooms I think you are talking about. Once again I am not sure about prices, so I don't want to tell you anything wrong :)
Have fun in Greece by the way!