Looking for an old anime I saw on Animax as a kid
This sounded like the novel A Little Princess to me, so I searched to see if there was an anime based on that. Turns out there is one called Princess Sarah, and apart from the green dress, it seems to match your description. (The protagonist does wear a green-ish dress in some of the pictures I've seen from the anime while searching for this, but she doesn't in a couple of them.) I am also not sure if the anime series aired at around the right time.