"It's fun to sing at Karaoke."
I would not say "sing at Karaoke", but "sing Karaoke" or "sing at Karaoke-night". Maybe they omit night as per the latter and think that is enough, the borrowed words tends to be used differently depending on region. Thanks for a great explanation on Japanese use of the word sing above!
Indicating a place where you can sing karaoke. In Japan they have "karaoke boxes" - I have never been myself, but I have heard friends talking about them. My guess from what friends have said is that they're small private rooms where you can sing karaoke and record yourself and your friends.