Etymology: Old English bí (big) accented; bĭ, be unaccented, = Old Frisian, Old Saxon, Middle Dutch, bî, be, (Dutch bij, be-), Old High German bî, bi, bĭ- (Middle High German bî, be-, German bei, be-), Gothic bi, bi- ‘about, by’ < Old Germanic *bi, probably cognate with Latin am-bi- prefix, Greek ἀμϕί, prep. and prefix ‘about’. (For the disappearance of am- in Germanic, compare Old Germanic bo-, with Latin am-bo-, Greek ἀμ-ϕο- both.)
Originally an adverbial particle of place; when prefixed to a verb it generally coalesced with the latter, and was treated as a prefix; when construed with substantives (in the dative or accusative, according as the relation was that of being near, or moving near to), it became, like other adverbs, a preposition. Compare the series: ‘þæt folc bí stód (bi-stód)’, ‘þæt folc him bí stod (him bi-stód)’, ‘þæt folc stód him bí,’ ‘the folk stood by him’, and the mod.English, ‘to stand by, stand by him, be a bystander’.
The single form bi of Old Germanic was subsequently, under the influence of the stress, differentiated into the strong or accented bî, bī (by, bij, bei), and the weak or stressless bĭ, later bĕ. The strong form was used for the adverb, the accented prefix of nouns, and a stressed preposition; the weak form for the stressless prefix of verbs, and a stressless preposition.
The influence of levelling, however, tended at length to make bî (by, etc.) the separate form in all cases, and to leave be- as the weak prefix; thus, while in Old English the prep. was both be and bi, in Middle English it was usually written bi, by, and modern English makes the preposition, like the adverb, by, in all positions and senses, and has be- only as a stressless prefix.
The same is true of modern German bei, be-, and Dutch bij, be-. But in pronunciation there was a weak and a strong form in Middle English (compare forms like be-sides, be-times, bum troth, bum Lady, byrlady), as is still usual in the dialects. In modern Scots bĕ is the ordinary form of the preposition unaccented, or in a weakened sense, as in ‘sit be the fire’, ‘written be a clerk’, ‘ane be ane’, by the form of the adverb and strong preposition, as in ‘stand by’, ‘to pass by a place be the railway’. This use of be as preposition has been uniform in the northern dialect since the earliest preserved Middle English specimens
The first recorded use of the preposition is in 898 AD, and they note that in 1000 it was first used in reference to time. The use which you asked about, as in "by this date", appeared in the 1300s: