This sounds like a question for Inok Nikolai!!
If the Russian Orthodox Church is similar to the Western Churches on this point, there will be a cycle of prayers and Biblical readings for each saint's day and other important days. In the Church of England, the prayers are called Collects for the Day. The Collects are printed in the Book of Common Prayer, along with the readings. For example, today is St Nicholas's Day, so there will be particular prayers and readings for that.
Given that the Imperial Family were very religious, and followed conventional Orthodox practices, I imagine they will have done something along these lines.
Ann