The oldest known famine, that I am aware of, was recorded by Herodotus. He wrote that

When Atys son of Manes was king, there occurred a severe famine throughout all of Lydia. For some time the Lydians tenaciously endured, but then, when they saw no end to it, they sought remedies, and different people came up with various solutions...On one day they played games so that they did not crave food; on the next, they stopped playing and ate, and they lived this way for eighteen years...(Histories 1.94.5)

Lydia was what is now western Turkey, and King Atys ruled sometime c. 1200 BC (if he was not merely myth or a fiction made by the later Mermnad dynasty; see Lancellotti, Attis, Between Myth and History, ch. 2). Herodotus goes on to describe the division of the people under Atys into two groups by lot, one half to remain, the other to emigrate in order to alleviate the suffering from famine.

Now, the above account is as much political creation myth as it is history, so I'll give a famine that is more likely to have occurred, and is the earliest recorded one for a particular area. In 60/61 CE, Prasutagus, the ruler of a tribe of Britains called Iceni near present day Norfolk, died. He may have been one of a number of unnamed kings who submitted to Emperor Claudius, helping ally his tribe with the Romans. Whether he made official submission or not, the Iceni paid tribute to the Romans, and conquered in 47 CE. After his death, Prasutagus intended for Nero and his daughters to inherit his kingdom, but the Romans "plundered by centurions, his house by slaves, as if they were the spoils of war" (Tacitus, Annals Bk. 14, c. 31). The soldiers beat the Prasutagus' daughter Boudicca, and raped Boudicca's daughters. At the same time as the raid, the procurator of Britain, Decianus Catus, chose to recall the loans made to pro-Roman Britons (ODNB Online, "Boudicca" by T. W. Potter). All these events provoked Boudicca to lead a rebellion against the Romans.

Tacitus tells us that Boudicca and her soldiers killed tens of thousands of Romans (Mattingly, Imperial Possession, pp. 111-112; the numbers given of 70-80,000 are an exaggeration, but perhaps as many as 40,000 did die). In response, Tacitus claims that " two thousand legionaries, eight cohorts of auxiliaries, and a thousand cavalry" were sent from the continent to join with the legionary soldiers remaining in Britain. After the arrival of the Roman army, Tacitus explains the destruction as:

The allied infantry and cavalry were placed in new winter quarters, and whatever tribes still wavered or were hostile were ravaged with fire and sword. Nothing however distressed the enemy so much as famine, for they had been careless about sowing corn, people of every age having gone to the war, while they reckoned on our supplies as their own. (Annals Bk. 14, c. 38).

Although Tacitus blames the Britons for not planting, the new Roman army seizing food stores likely contributed to the shortage of food. This British famine was the first recorded in written histories, as far as I am aware of. Unlike the Lydian famine, the famine has a near contemporary author in Tacitus, and his father Gnaeus Julius Agricola, was governor of Britain CE 78–84, and Tacitus served on his fathers staff.