Aleinu originally is a part of the Rosh HaShanah (Musaf) liturgy. It is the introduction to the verses of Malkhiyyot (Divine Sovereignty) which, together with the verses of Zikhronot (Divine Remembrance/Providential Attentiveness) and Shofarot (Shofar Blasts), are unique to this liturgy. Because this is a major theme of the day and because one particularly stands in God's sovereign presence on this day, it became customary to open the ark for this paragraph (through Va'anahnu).
The congregation rises and, in some places, would actually perform a full prostration at Va'anahnu (nowadays it is more usual in traditional synagogues for only the service leader to do this; it is not done at all in most Reform synagogues). This is what sometimes is referred to as "The Great Aleinu." A special melody is used in this context. In the early medieval liturgies, the Aleinu paragraph was also recited in the same location, in the Musaf service, on Yom Kippur, at the beginning of Seder Ha'avodah, the commemoration of the Temple Service on the Day of Atonement. Once again, the ark would be opened, and the congregation would stand and prostrate themselves.
In the thirteenth century, in the Rhineland, the Aleinu paragraphs (including two of the ten Malkhiyyot verses, one of which is Bayom hahu) were taken over into the daily services--weekdays and Shabbatot, as a fitting conclusion for every service, since they articulate the messianic hope for the ultimate acknowledgement by all humans of the sovereignty of Israel's God. While the congregation always stands for Aleinu at daily services, the ark is never opened; that gesture is reserved only for the High Holy Days.
The Reform custom of opening the ark for Aleinu on weekdays and Sabbaths is not universal, but does derive from the days of "classical" Reform; it is "borrowed" from High Holy Day practice, where it is more appropriate. You are correct that it is not indicated in the UPB or any subsequent Reform prayer book. There the traditional custom persists of opening the ark only for the "Great Aleynu," not the "regular" one. So, strictly speaking, the ark should not be opened before the Aleinu throughout the year, only on the High Holy Days. But the congregation always rises to recite this text. Rick