これであっていますか？

これもOK:
◎　When I got home, the TV program had already started.

△　When I arrived (at) home, the TV program had already started.
◎　When I came home, the TV program had already started.

ご質問の文脈で"arrived (at) home"と表現するのは不自然であるが、"came home"であれば問題なしという解説が次のサイトで示されています:

Which one is correct, "arrive at home," "arrived home," "arrive at my home?."
https://josecarilloforum.com/forum/index.php?topic=326.0