This isn't about kings necessarily, but for emperors, yes the option was absolutely open to sit in a "throne room" all day, if they wanted. The emperor Claudius was somewhat infamous for this, his general persona was that of a Republican magistrate and he was perhaps the only one of the emperors to really take his Republican offices seriously. He had a habit of spending all day on court cases, and was an extremely energetic censor. Ancient authors were somewhat divided on this: Suetonius generally notes with approval his administrative reforms in a single short section (14), and then goes on in great detail criticizing the particularities of his judgments. It is pretty funny too, by the way, particularly if you are familiar with the general literary culture of the time you can see how it becomes a parody of Claudius' pretensions. Whether you view the disparity of Suetonius' focus as an example of an elaborate critical device, presenting a staid "official" account which he then gleefully undercuts with lengthy descriptions of details in a delightful subversion of the propaganda of the imperial court, or simply because stories about Claudius being kind of muddle headed and socially inept are funnier than ones about changes in court procedure, largely comes down to how you view Suetonius. Tacitus more clearly has a subversive purpose, as he presents a pretty stark juxtaposition of Claudius' wife running wanton with half the senators and conspiring to murder the other half, the emperor himself doing some damned thing with the alphabet. To quote:

She had grown so frantically enamoured of Caius Silius, the handsomest of the young nobility of Rome, that she drove from his bed Junia Silana, a high-born lady, and had her lover wholly to herself. Silius was not unconscious of his wickedness and his peril; but a refusal would have insured destruction, and he had some hope of escaping exposure; the prize too was great, so he consoled himself by awaiting the future and enjoying the present. As for her, careless of concealment, she went continually with a numerous retinue to his house, she haunted his steps, showered on him wealth and honours, and, at last, as though empire had passed to another, the slaves, the freedmen, the very furniture of the emperor were to be seen in the possession of the paramour.

Claudius meanwhile, who knew nothing about his wife, and was busy with his functions as censor, published edicts severely rebuking the lawlessness of the people in the theatre, when they insulted Caius Pomponius, an ex-consul, who furnished verses for the stage, and certain ladies of rank. He introduced too a law restraining the cruel greed of the usurers, and forbidding them to lend at interest sums repayable on a father's death. He also conveyed by an aqueduct into Rome the waters which flow from the hills of Simbrua. And he likewise invented and published for use some new letters, having discovered, as he said, that even the Greek alphabet had not been completed at once.

Anyway, that is a somewhat roundabout answer to your second question, did they have anything better to do, as the answer was, for many people, yeah.

As for what they would do, a lot of what Claudius did was hear cases. The Roman empire had an overlapping set of legal systems, with most places maintaining their traditional codes and courts. But it also had a great deal of Roman colonists and citizens scattered around the place, and citizens had the option of appeal all the way to Rome. This meant that not only was Claudius hearing legal disputes from the city and surrounding area, but theoretically throughout the empire. Add to this, he might also be hashing over details of administration, drawing up plans for construction projects, hearing petitions from all across the empire, and you can understand how he might be kept busy in a place for an entire day.

But of course emperors did have the option to not do this. The emperor Tiberius for example "retired" to a villa on Capri, and many other went on campaign and the like. But for the few that really had the interest in detailed governing, the option was certainly there.