Why should “be” come after “neither a borrower nor lender,” not before them?

Solution 1:

Briefly, Polonius inverts the order because it's Elizabethan blank verse and he can. He is to all appearances* a pompous blowhard who loves to hear himself speak. In fact, at one point he gets so caught up in his legal rhetoric and endless qualifications and syntactic inversions—in short, in the sound of his own voice—that he completely loses track of what he's saying:

POL:  Wherefore should you do this?
REY:                Ay, my lord,
    I would know that.
POL:           Marry, sir, here's my drift;
    And I believe it is a fetch of wit:
    You laying these slight sullies on my son,
    As ‘twere a thing a little soil’d i’ the working,
    Mark you,
    Your party in converse, him you would sound,
    Having ever seen in the prenominate crimes
    The youth you breathe of guilty, be assured
    He closes with you in this consequence;
    ‘Good sir,’ or so, or ‘friend,’ or ‘gentleman,’
    According to the phrase or the addition
    Of man and country.
REY:         Very good, my lord.
POL:  And then, sir, does a’ this—a’ does
    —What was I about to say? By the mass, I was about
     to say something: where did I leave?
REY:  At ‘closes in the consequence,’ at ‘friend or so,’ and ‘gentleman.’
POL:  At ‘closes in the consequence,’ ay, marry;
    He closes thus: ‘I know the gentleman;
    I saw him yesterday, or t’other day,
    Or then, or then; with such, or such; and, as you say,
    There was a’ gaming; there o’ertook in’s rouse;
    There falling out at tennis:' or perchance,
    ‘I saw him enter such a house of sale,’
    Videlicet, a brothel, or so forth.
    See you now:
    Your bait of falsehood takes this carp of truth:
    And thus do we of wisdom and of reach,
    With windlasses and with assays of bias,
    By indirections find directions out:
    So by my former lecture and advice,
    Shall you my son.

* I happen to believe that Polonius is a devious knave pretending to be a fool; but that’s LitCrit.

Solution 2:

This is an example of inverse copular construction, a kind of subject-verb inversion that blurs the difference between subject and predicate. It's somewhat unusual because the imperative mood removes the subject you from the sentence, but it still has the same basic form and the same function: changing the emphasis, tone, or register of the sentence.

In this case, the inversion indicates a proverbial register: Polonius giving sage advice to his departing son. In the original source, Polonius is a gasbag, so the "sage" advice is ironic, but many people repeat it at face value, as a true proverb.