update: The oldest manuscripts of this prophecy do not mention airplanes or automobiles, according to an email I received from Thomas Anton Reuter, who wrote an article on this subject. (This could concievably be confirmed by a Javanese speaker who clicks the Google Books link below) Reuter stresses that he does not have every single source of the prophecy, but as of yet, there is no written evidence of this prediction that predates the actual appearance of airplanes and automobiles in Indonesia.
The best English-language source for the Jayabaya prophecy is Thomas Anton Reuter, “The Once and Future King: Utopianism as Political Practice in Indonesia”. In Utopía: 500 años, ed. Pablo Guerra (Bogotá: Ediciones Universidad Cooperativa de Colombia, 2016).
The Jayabaya prophecies specifcally predict a coming crisis of epic
proportions, which the text refers to as the ‘time of madness’ (jaman
edan). For the majority of Javanese the description of this crisis, as
provided in the prophecy texts (for example, that women will wear
men’s clothing, and young people will cease to respect their elders),
closely resembles actual societal conditions of the present day. This
state of chaotic disorder and moral decay is predicted to culminate in
a major calamity, followed by a ‘return of the king’ and a restoration
of cosmic harmony in the form of a utopian society. At various times,
certain individuals have proclaimed themselves as the new ratu adil,
and a few of them gained some recognition in connection with 19th
century rebellions against Dutch colonialism.
He quotes a modern oral retelling of the prophecy:
When carriages drive without horses, ships ﬂy through the sky, and a
necklace of iron surrounds the island of Java; When women wear men’s
clothing and children neglect their aged parents, know that the time
of madness has begun.
(From the Ramalan Jayabaya, Oral transmission,
Here I should note that the general idea of a "time of madness" parallels the actual ancient prophecies found in the Mahābhārata which date to 400 BC or earlier:
And all men towards the end of the Yuga will become members of one common order, without distinction of any kind. And sires will not forgive sons, and sons will not forgive sires. And when the end approaches, wives will not wait upon and serve their husbands. (etc. etc.)
The Mahābhārata came to Indonesia around the year 1000 and was already a cultural staple by the time Jayabaya came to power over 100 years later. So, surprisingly enough, although we have no manuscripts of "Jayabaya's prophecy" from before the 19th century, it is easy to tell that the prophecy is much older than this, and it's quite possible that the prophecy in its initial form dated to the 12th century when oral retellings of the Mahābhārata may have included these Kali Yuga prophecies. Reuter writes,
In the foreword to the 1835 edition the scribe complains bitterly that
his predecessor has ‘Islamised’ the ancient narrative, warning that
such a shamefully manipulation of sacred knowledge can lead to
dangerous personal consequences for the scribe and will confuse the
public with false interpretations. It is not diffcult to see what the
concerned scribe is referring to. For example, the text names the
Sultan of ‘Rum’ (Constantinople / Istanbul) as the one who initiated
the original human settlement of Java rather than the Hindu saint Aji
Saka, who is commonly named as the founder of civilization in Javanese
folk tradition and in classical Javanese literature. A scribe in a
royal library would have been well aware that this modifcation of the
text was a conscious act of historical revisionism.
18th century Islamized text by and large still follows the plot of an
earlier, Hindu version, as is evident from what fragments of the
original Hindu version can still be found today in other manuscripts.
Reuter cites an 1848 print edition of the prophecy by a Dutch colonist, which is happily available on Google Books. But alas, this is written entirely in Javanese script which I found very hard to read. Reuter does not discuss the content of this 1848 edition or whether it mentions automobiles or "flying ships."
A Google Books search turned up another reference, in a Dutch-language book from 1892 (Jan Laurens Andries Brandes, Een Verslag ... eene Verzameling Javaansche en Madoereesche handschriften), to another manuscript of the Jayabaya prophecy dating to between 1650 and 1715 AD, which was located in the Royal Batavian Society of Arts and Sciences at the time of writing. Reuter may have missed this one, or the manuscript may have deteriorated. The book does not describe the prophecy in detail other than to mention that it prophecies "diseases and terror and famine" and to link it with peasant uprisings.
Due to lack of well-preserved manuscripts, we cannot answer the question of whether "Jayabaya predicted cars and airplanes" but it's possible that there was a mention of "flying ships" in an oral tradition predating the first airplanes. As many people have commented, this could be interpreted as the poet's flight of fancy, etc.