Hosted feature services don't currently support time. It's something we want to support and will release when available. So, there is no hosted way to support time aware layers. You need to have ArcGIS Server currently to do this.
When you "share as" from ArcMap, you get a tiled map service by default. You need to explicitly enable the feature service capability (and optionally turn off tiles if you don't want them).
Thanks,
Mike