Construction of “woe is me”

It is indeed old, and can be found in Beowulf:

Wa bið þæm þe sceal þurh sliðne nið sawle bescufan in fyres fæþm, frofre ne wenan, wihte gewendan; wel bið þæm þe mot æfter deaðdæge drihten secean ond to fæder fæþmum freoðo wilnian.

Woe be to him who through severe affliction thrust his soul into the fire’s embrace, hope not for relief, or to change at all; Well be to him who after his death may seek the Lord and long for peace in the Father’s arms.

_{“Woe to the Rich and the Sordid Fellows”: The Syntax, Semantics and Information Structure of ‘Woe’ in Indo-European}

Note the emphasis (which is from the cited paper). Woe here is parallel to well; because well is an adverb, so is woe.

(Note: In this particular sense, "Wa bið þæm" is closest in meaning to "Cursed is he".)

The OED explains that this usage came from the interjection woe (which was a "common Indo-European interjection" also used in OE):

Arising as an adverbial use of the interjection with the dative, although in later use probably often interpreted as a noun

To some extent, English has (in many cases) "before-the-verb and after-the-verb constructs. "Who's that at the door?" is usually answered by "It's me," rather than "It is I." "Me" (much more than "he" or "she") behaves a bit like the French "moi."

Also "he," "she," and "me," rhyme for whatever that's worth.