I will try my best but it will be a very big generalization. First I would like to mention Rome because you mentionned specifically for the king so am obliged to mention briefly Roman kingship. As the king of Rome was ruling a single city, even if it was arguably already one of the most powerful in the area with a relatively limited territory...I would say comparatively to the rest, well pretty high, especially if you are serving in the army that he was supposed to command or during religious ceremonies that he was supposed to perform. There should also be interactions on the political level but I really don't want to enter into technicalities here. In addition, considering that the Circus Maximus is an extremely old racing field and that racing tradition was already strong in the other cities of the area (known through potteries depicing races), it would be logical to assume that the chief of the city would be involve in such manifestations

Of course what you really meant was the roman emperors...actually it's not that different. As politics was tightly controlled and fundamentally the popular assembly lost influence and interest the main occasion would still be public religious ceremonies...in this I include the triumph which is in a way religious and martial and of course games. In a way you can see that the main ways the ruling power would show itself to the public are quite ritualized. But, in the years the affirmations of Augustus regime, there was no imperial palace and the text tell us about honours granted to him through popular movement, massing in front of his home. But the roman emperor is also someone who is away and sometimes for a long time. Away from Rome but also when in Rome, away from everywhere else in the Empire.

Now the real problem is that such common interractions are not what ancient historians are interested in. Or more precisely in front of the ruler there is "the people" a mass that has from main caracteristics to be unidentifiable.

You wanted a word about Alexander the Great... It's fundamentally worse. As a king constantly on the move conquering new lands lots of his new subjects would not have the opportunity to see him. On the contrary the fact that he spent 8 years campaigning with the same troops (even if he received reinforcements multiple times) certainly created a strong bond between him and his men but again he was still surrounded by his loyal officers/bodyguards and the sources don't tell much about the others. Nevertheless the fact that addressed his troops directly (even if the texts we have are reconstruction) show that in a sense it was a personnal relationship.