How did Ducat come to mean ticket?

Solution 1:

Its current meaning as ticket probably derives from its prominent usage in Shakespeare's "The Merchant of Venice:"

Ducat:

• A ticket or pass to a show, game, race, etc (1874+)

(The Dictionary of American Slang)

Etymology:

• So called for the name or effigy of Roger II of Sicily, Duke of Apulia, which first issued the coins (c.1140). Byzantine emperor Constantine X had the Greek form doux struck on his coins during his reign (1059-1067). Over the years it was a unit of currency of varying value in Holland, Russia, Austria, Sweden, Venice, etc.

• Remained popular in slang for "money" or "ticket" from its prominence in "The Merchant of Venice."

(Etymonline)

William Shakespeare The Merchant of Venice:

• Shylock: "My daughter! O my ducats! Fled with a Christian! O my Christian ducats! Justice! the law! my ducats, and my daughter!..."

Solution 2:

J.E. Lighter, Random House Historical Dictionary of American Slang (1994) identifies two slang meanings of ducat, the first of which appears to be almost a century older than the second:

ducat n. 1. a dollar; (pl.) money {In 1775 quot., simply 'a piece of money'.} [Earliest cited examples:] {1775 R.B. Sheridan, in OED: I shall be entitled to the girl's fortune, without settling a ducat upon her.} 1866 [Henry] Williams, Gay Life in N[ew] Y[ork! Or, Fast Men and Grass Widows] 88: My money is good and I'll bet my "ducats" he can take any of the crowd agin him, if they don't double-team him. 1873 Hotten Slang Dict[ionary] 152: Ducats, money.—Theatrical slang. 1878 Nat[ional] Police Gaz[ette] (Apr. 24) 6: I fight you for fun or ducats.

2. a ticket, esp. for transportation or admission. [Earliest cited examples:] 1871 Banka Prison Life 493: Railroad Ticket...Ducket. 1873 Hotten Slang Dict[ionary]: Ducket, a ticket of any kind. Usually applied to pawnbroker's duplicates and raffle-cards. 1879 Macmillan's Mag[azine] (Oct.) 501: I took a ducket for Sutton in Surrey. 1910 N[ew] Y[ork] Eve[ning] Jour[nal] (May 28) 11: I've got two ducats for th' fight.

The Sheridan reference occurs in his comic opera The Duenna (1775).

In John Hotten, The Slang Dictionary, the entry for ducats as theatrical slang for money first appears in the 1864 edition. Hotten (a Londoner) includes an entry for ducket in the 1874 edition of his dictionary (and presumably in an 1873 edition not available online) and includes with it a suggested etymological source:

Ducket, a ticket of any kind. Generally applied to pawnbroker's duplicates and raffle-cards. Probably from DOCKET.

But if duckets originally applied to a "pawnbroker's duplicates," I'm surprised that Hotten didn't suggest that it might be a corruption of duplicate or perhaps a portmanteau of duplicate ticket.

Meanwhile, the theory that ducats as slang for money is traceable specifically to The Merchant of Venice (as noted in Josh's answer) goes back at least to J.S. Farmer & W.E. Henley, Slang and Its Analogues, Past and Present, volume 2 (1891):

DUCATS, subs. (theatrical). — 1. Money. {Probably from Shylock and The Merchant of Venice.} ... [Citation:] 1853. Wh. Melvile, Digby Grand, ch. vi. The Jews have always appeared to me a calumniated race. From spendthrift King John downwards, the Christian has ever pocketed the DUCATS and abused the donor.

2. (thieves'). — Specifically a railway ticket; also a pawnbroker's duplicate; raffle-card, or BRIEF [card "tampered with for the purpose of swindling"]. Also DUCKET. [Citation:] 1879. J.W. Horsley, in Macm[illan's] Mag[azine] xl, 501. So I took a DUCAT (ticket) for Sutton in Surrey.

Lighter's earliest citation for ducket as "ticket" is from J. Harrie Banka, State Prison Life: By One Who Has Been There, in a very brief glossary that includes the entry quoted (in full) by Lighter. The interesting thing here is that Banda is an American whereas Hotten (publishing just two years later) is British. The upshot is that, as of 1873, ducket was transatlantic slang for "ticket."

One early instance of ducket in this sense comes from "Slang of Petty Thieves: A Storehouse from Which Everyday Speech Is Replenished," in the Indianapolis [Indiana] Leader (July 9, 1881), reprinted from the New York Sun, in which a detective explains the criminal cant of the time:

" ... A 'rattler' is a train of cars, a 'ducket' is a railroad ticket, a 'tip' is the railroad office, a 'prod' means a horse, a 'drag' is a wagon, and 'pinched' or 'nipped' means arrested."

More specific is the entry for ducat from "Fonetic Spelling: Inquirers Want to Know What Certain Words Mean," in the [Grand Raids, Wisconsin] Wood County Reporter (October 19, 1899):

DUCAT, n. A ticket of admission to a race track or other inclosure; the goods to get by the boy on the gate

The specific association of ducket with "pawn ticket" reappears in "Latest Dictionary of Slang: Underworld Vocabulary Undergoes Constant Changes—Source with Gypsy Tribes," in the [Honolulu, Hawaii] Commercial Advertiser (August 24, 27, 1904), which attributes the glossary that makes up much of the story to "'A strong-arm' man now serving ten years' sentence in Sing-Sing [prison in New York]":

Ducket—A pawn ticket.

Conclusion

The consensus among generally sound authorities—including Jonathon Green, Chambers Slang Dictionary (2008) and Barbara Kipfer & Robert Chapman, Dictionary of American Slang, fourth edition (2007—cited in Josh's answer) is that ducat in the sense of "ticket" comes from ducat in the sense of "money," which is itself traceable to Italian coins of the sixteenth century. John Ayto, The Oxford Dictionary of Slang (1999) generally agrees but also revives the notion from Hotten that the word docket may be involved as well:

ducat, ducket(t) (1871) Applied to a a ticket of admission; probably from earlier sense, coin; perhaps influenced by docket and ticket

This is a reasonable view. But to me the sequence of events from 1864 through 1873 is sufficiently strange to raise some doubts about the simple progression that most slang dictionaries trace, from ducats (Italian coins) to ducats (money, figuratively) to duckets (tickets). Here is the timeline:

1864 John Hotten adds ducats as theatrical slang for "money" to his slang dictionary. (It doesn't appear in the 1860 second edition of the dictionary.)

1866 Henry Williams, in a book about life in New York, uses ducats in the sense of money, indicating that the usage exists in England and the United States.

1871 J. Harrie Banka cites ducats in the sense of "tickets" as U.S. prison slang.

1873 John Hotten adds ducket in the sense of "ticket" to his dictionary, indicating that the usage exists in England and the United States—and concludes that it probably derives from docket.

Hotten was not a professional linguist, but he had the advantage of being on hand during the period when ducket as "ticket" achieved enough prominence in popular usage to justify his including it in his slang dictionary. The fact that Hotten shows no inclination to connect the new slang term to the earlier term ducats—a word that sits two entries away from ducket in the 1874 edition of The Slang Dictionary—makes me wonder why he didn't see any connection between the two.