Phones should be programmed to accept or reject calls by the owner, because it is a really dumb idea to be able to wake people up knowing only their phone number. Notably, this is already the case: cellphones have mute/vibrate switches, and if you have a smart phone you probably have various "do not disturb" modes that do more advanced routing.
As a followup and to actually resolve the question for the call placer, phones could broadcast their availability status and this status could be queryable by phone number.
Stardates have their own problems but I'm not convinced those problems are harder than the problems we have with timezones now. We've built up a terrible amount of infrastructure around timezone handling, we should not be scared off when stardates require unique infrastructure projects.