The questions asks,
Or would it have just advertised a new comedy by Shakespeare?
Although no play-bills have survived from the English theatre of the sixteenth and seventeenth centuries, we have a letter from poet laureate and playright John Dryden to his cousin, containing an observation on the practice of naming the playwright in the advertisement, which he says was an innovation in the late 17th century:
This Day was playd a reviv’d Comedy of Mr Congreve’s calld the Double Dealer, which was never very takeing; in the play bill was printed,—Written by Mr Congreve; with Severall Expressions omitted: What kind of Expressions those were you may easily ghess; if you have seen the Monday’s Gazette, wherein is the Kings Order, for the reformation of the Stage: but the printing an Authours name, in a Play bill, is a new manner of proceeding, at least in England.
John Dryden (4th March 1699). Letter to Elizabeth Steward. In Charles E. Ward, ed. (1942). The Letters of John Dryden, with Letters Addressed to Him, pp. 112–113. Duke University Press.
If Dryden was right about this, then it is doubtful whether Shakespeare’s name appeared on the play-bills for his plays. Most likely (as with films today) it was the actors who drew the crowds and not the writers.
Some indication of the attitude of theatre companies to their publicists can be gathered from The Transproser Rehears’d (1673) by Richard Leigh, a poet and an actor in the Duke of York’s theatre. This pamphlet was an attack upon Andrew Marvell’s The Rehearsal Transpros’d (1672), and when Leigh needed a devastating insult for Marvell, the first thing that came to mind was to portray the latter as a writer of play-bills and other advertisements.
The Author† of the Animadversions upon the Preface to Bishop Bramhalls Vindication, &c.‡ (if it be not too great a favour to call him an Author that writes a Book upon a Preface) having potted up a Play-Bill for the Title of his Book: And here by the way, we cannot but congratulate his honourable employ, and question not but to hear of his being prefer’d from writing of Bills for the Play-houses to penning of Advertisements for the Stage-Coaches and Bills for the Pox, and after a proficiency therein, to be admitted upon the next vacancy, to form Draughts for the Arithmetick and Shorthand-men, and frame Tickets for the Rope-dancers and the Royall-Sport of Cock-fighting, that so he may arrive in a short time to be Author of most of those ingenious Labours which curious Readers admire at Pissing times in their passage between White-hall and Temple-bar.
Richard Leigh (1673). The Transproser Rehears’d, pp. 1–2. Oxford: assigns of Hugo Grotius and Jacob Van Harmine.
† That is, Marvell. ‡ That is, The Rehearsal Transpros’d.