Hi, it's an indoor tropical so don't worry about cold dormancy, per se, though it is quite likely to have a few weeks in the later fall where it stops growing and won't need watering quite as often, so be careful about that (take your cue from the soil, not a schedule). I think it may either be a Serissa or some kind of Ficus, not always easy to tell from a picture, but if leaves are on the stiffer side vs soft and easily 'rollable', plus if anyone suggested it gets white flowers, it would be a serissa.

Eugenia myrtifolia . . . aka brush cherry

Al