While it would vary by an Emperor's temperament, age, etc Byzantine Emperors were usually expected to do certain tasks on a daily basis. This usually included prayer in either the palace chapel or if a major holiday, the Hagia Sophia. Partaking in religious processions was not uncommon for an Emperor and often a great PR move with Constantinople's residents.

Emperors would preside over the chariot races in the hippodrome and some Emperor's (Constantine VIII (r. 1025-8), in particular) would have favourite drivers they would shower gifts upon.

Cleanliness was very important to an Emperor and time spent at a bath was a daily, often more than once daily, activity. Byzantine baths were less like our modern bathtubs and closer to swimming pools, and a bath facility usually had a hot and cold bath. Romanus III (r. 1028-34) was a avid bather and was in fact assassinated while taking a bath.

Emperors had other hobbies that their position allowed them, one in vogue in the 11th century was having large public and private gardens. Constantine IX (r.1042-55) doted on his gardens quite a bit.

As for actual duties, the Emperor would normally receive the usual dignitaries and petitions in his throne room. If his guest were from less 'civilized' parts of the world, the Emperor would often terrify them with his steam pressurized throne that could raise itself to great heights at a touch of switch. Or he could dazzle them with singing mechanical birds or mechanical lions that could roar. For more respected visitors the Emperor often kept up a silent, stoic demeanour, speaking only when necessary and letting his advisors/palace staff do most of the leg work. This is not to say Emperors weren't sociable, often in private they were very personable, but in public they had an image of austere magnanimity to upkeep.

Emperors were often military men and would be away on campaign. Emperors such as Basil II (r. 976-1025) or Alexius I (r. 1081-1118) could be away for months at a time on campaign and the normal protocols in Constantinople would be performed by a subordinate or co-emperor.

Emperors were sometimes learned men and had a large group of scholars, philosophers and the like around them. Large parts of the day could be spent chatting with them. Constantine IX went as far as opening the University of Constantinople in 1045 and putting friends like the historian Michael Psellus in charge.

Meals were usually eaten in private but some Emperors were known for holding large banquets with all sorts of entertainment, church approved or not (from jugglers and dancers to wrestlers and poetry reading). Constantine VIII and Romanus III seemed to particularly to enjoy such things.

The Emperors days were jammed full of these kind of events and one needs to remember the many religious holidays, fasts, feasts, etc that the Emperor had to preside over. Almost every day was devoted to a particular saint or saints. An Emperor was expected to be at the front of the church and be seen by all the important VIP's in the capital.

On top of this, the Emperor had to spend time with his family and the likelihood was he had one or more mistresses to spend time with.

I hope this is a good start!

SOURCES: Everyday life in Byzantium by Tamara Talbot Rice; Byzantine Civilization by Steven Runciman

the Emperor would often terrify them with his steam pressurized throne that could raise itself to great heights at a touch of switch.

This is pretty interesting. Where can I read about this?

I'm guessing this is your homework assignment?