Actually an absolute monarchy feudal system makes a lot of sense for a vast civilization. (There are very much major problems with democracy, namely it is hard enough to get twenty people in a club of shared interests to fully agree on something, let alone a few hundred billion spanning dozens of planets.)
At that scale the emperor doesn't need to make direct decisions, he handles high level elements and delegates the actual work and details to others. The emperor is the one setting goals, others do the work to actually achieve them. The key point of the emperor is then to basically bully everyone into making sure the job gets done, and ensuring the right people are in the right places at the highest level. He becomes who the lesser people answer to when things go wrong.
Factories in one sector are under preforming, and that is impacting Imperial Economics? Emperor yells at his minster/lord/whatever in charge of that region, who in turn yells at his subordinates, who yell at their subordinates, etc, etc, and eventually the problem is fixed, or another round of yelling takes place and a few heads roll. If it continues with changes (that the emperor doesn't even really need to know about) at the lower level not having an effect, then the emperor 'steps in' and puts someone else in charge of the whole mess.
This isn't true. Feudal systems are useless at large population values and they are not good for any sort of ingenuity based society. Heredity is a fucking terrible system for quality rulers. Its rife with corruption, too.
Aside from Dune which was more about planet ecology than interstellar politics there are almost no examples of imperial systems in speculative fiction. Sometimes you have them in role playing IPs but that's because its "cool" and "epic."
Well to answer the title directly, nothing. Because the kind of culture needed to run an interstellar society is as far from imperial as can be. Even aside from the fact that a single person could never handle that much complexity. I am speaking strictly to feasibility of space empires.
As far as what random crap you could make up for him to do, he probably does whatever regular emperors do, because otherwise why call him an emperor?
Lots of wining and dining, orgies, random executions, parades in dress cloths. If you have instant communication he might be an admiral of a fleet or something.
From the games I've seen of yours you aren't seriously asking what his day to day routine is. Day to day routine is boring as shit whether you are an emperor or a serf.
He would probably have meetings with his spymaster and treasurer, but in my experience video games tend to abstract that out to always on menus like resource bars.