What I've heard so far:
The Asian ideas of impermanence in Buddhism and/or Hinduism
An Old English poem, titled (I believe) The Lament of Scol(?), in which each stanza is about some mythical character's horrific troubles, ending each one with something along the lines of "That passed, so too will this".
A Persian story about a king (I think) who wanted a ring that would magically make you happy when sad, and sad when happy, and wise men gifted him with a ring that had the phrase inscribed on it
A group of similar Jewish stories surrounding King Solomon, similar to the Persian story, but with Solomon as the giver or receiver of the ring -- I believe I've seen this story attributed to Maimonides, but I'm not sure. The Jewish stories also had the unique facet that some of them have the ring only bearing the first letter of each of the three words in the phrase, rather than the whole thing.
My thoughts are that the Asian schools of thought are too distant to be considered related, and I find it unlikely that something in Jewish and Persian thought (as well as Turkish, but I haven't found any attribution to Turkish sources, only that the phrase is known there as well) would have a parallel that is genetically^1 related in Anglo-Saxon England. I find it fairly likely that the Persian and Jewish stories, however, are genetically related, since the stories are very similar, and the groups have historically been in closer proximity than England and the Middle East.
So, keeping this in mind, where did the phrase, and in turn the Persian/Jewish stories, come from?
^(1: When I say genetically related, I'm using it in the same sense as in linguistics, where something being genetically related means that either one descends from the other, or they both descend from the same source.)
It is in fact a Persian phrase: "In niz bogzarad". As you note, a similar phrase does appear in the 10th century poem "Deor's Lament": "Þæs ofereode,þisses swa mæg", "That was overcome, this so may", but this doesn't appear to be the origin of the modern usage, and it isn't really the same phrase (it is an active phrase calling for action, not a passive one to soothe or remind you of the temporality of feelings). Its popularity traces to the mid-19th century, where we find numerous references to a story about "a Sultan" or "an Eastern Sage" by people like Edward FitzGerald and indeed Abraham Lincoln:
It is said an Eastern monarch once charged his wise men to invent him a sentence, to be ever in view, and which should be true and appropriate in all times and situations. They presented him the words: "And this, too, shall pass away." How much it expresses! How chastening in the hour of pride! How consoling in the depths of affliction!
It's interesting to check using Google Ngrams that it first appears in the early 19th century (as far as its archive is concerned anyway). The earliest examples seem to be from various Christian magazines recounting sermons and the like; the earliest I can find is from 1824, credited to a nameless "Eastern sage" or "Prince in India" who had a ring furnished for "Adversity and prosperity alike".
The original story is frequently attributed to the Sufi sage Attar. Now, I would really, really like to trace down whether this story actually appears in one of Attar's published works, or whether it is simply a later attribution. Because I am unable to track down any reference to one of his own works, I am leaning toward the latter, that it is an apocryphal attribution, but because I cannot actually read his works for myself, I cannot confirm. I believe it is safe to say that its popularity in later lore derives from Persian sufi mystics, though. Its origins beyond that, if there are any, appear obscure and I could only speculate beyond that.