All right, so...why is "this" marked incorrect? Seems a little nitpicky to me.
"I was home at the time." - that would probably be the more common way to say it in English, (although "at that time" would technically work as well) and especially since there isn't a word for "the," that answer should be marked as correct, no?
Well, it depends on what we want to say: "It's eight o'clock now. At this time yesterday I was at home." "Where were you yesterday at eight p.m.? - I was (at) home at that/the time".
Accusative is used. Accusative case is quite common with time expressions. http://www.alphadictionary.com/rusgrammar/time.html
был is the past form for быть
It's masculine and singular (so the speaker should be male from this sentence)