During the entire runtime of Episode IV: A New Hope, even though he is mentioned a few times, the Emperor is never actually shown on screen. And even when he is mentioned it's more of a passing reference to the fact that he exists.
Any reason for this?
And did this have anything to do with Vader's original design as a Prince of some sorts?
In universe answer:
He is in charge of a freaking Galactic Empire, he's busy. Sure, he probably was there for the launching ceremony of the Death Star, but he would spend most of his time in the center of government. And over the week(or so) that the movie covers he is dissolving the Senate and replacing it with a structure of military governors and Moff's. So command of greatest military installation/vessel was given to the greatest Imperial commander (Grand Moff Tarkin) and the personal enforcer (Darth Vader). The only reason he was on the Second Death Star was to add a layer of bait to insure the Rebellion attack.
Behind the scenes answer:
James Sheridan's answer is wonderfully in depth.
Writer's answer: Royal Canadian Bandit goes in depth on the literary reasons not to show the Emperor openly until the end of the series. In many ways it is similar to the reason you rarely if ever see the shark in Jaws. Unfortunately it is something that was no longer forefront in Lucas's mind in the new trilogy, along with many other aspects of storytelling that he was steeped in early in his career.
The Emperor was originally intended to be a clever bureaucrat and politician by the name of Cos Dashit, later changed to Cos Palpatine, who became President/ Chancellor (the exact title was never made clear in the original drafts), proclaimed himself Emperor - perfectly legally - then left the day-to-day running of his Empire to his advisors, who were to be the real perpetrators of the heinous crimes committed by the Empire.
Darth Vader was originally conceived to be a character named Prince Valorum, and his design was slightly different, but otherwise he was essentially the same character. Tarkin was meant to be the leader of a crazy religious cult that worked with the Empire.
While Lucas has never commented on this publicly, to an historian's trained eye it seems that there were far more parallels to the Roman Empire and Augustus in the original drafts than the Nazis, which later became predominant. As such, the use of royal titles such as Prince, Princess, and Moff (which is apparently based on Turkish word for some kind of priest-soldier) made perfect sense; the Holy Roman Empire was more of a confederation of different principalities and bishoprics under the rule of a single Emperor, rather than a solid bloc as Nazi Germany was. Of course, Augustus had nothing to do with the later Holy Roman Empire, but I hardly expect historical accuracy from George Lucas.
EDIT: Wookieepedia finally decided to co-operate. Links added.
It's classic storytelling to gradually build up the main villain, and only reveal him or her near the end. The villain can be much more scary if his reputation is slowly developed over a period of time before the audience actually sees him. Meanwhile the heroes can fight his minions, led by the chief henchman/enforcer. (Warning: TVTropes links.)
In Episodes 4-6, the Emperor is just an ugly old man in a hooded robe who walks with a cane. By itself, that's not especially scary. But in ANH we don't see him at all, and in TESB he appears only briefly as a holographic projection. By the time the audience sees him "live" in ROTJ, he has been built up for two and a half movies.
Not only is he the iron-fisted tyrant who rules the galaxy and threatens to crush the Rebellion, he is Darth Vader's boss. Vader himself is afraid of the Emperor, as we know from his conversation with Luke in TESB. We have had plenty of time to see how scary Darth Vader is, so anyone who frightens Vader must be incredibly dangerous. When the audience finally meets the Emperor, they understand his feeble appearance conceals tremendous power and evil.
