Argeleb's father was Malvegil, during whose reign the Nazgûl reappeared in Middle-earth, and the realm of Angmar was founded in opposition to the fractured remnant of Arnor. When Argeleb succeeded his father, he sought to reunify the three kingdoms, since his own realm of Arthedain was the only one in which an heir of Isildur's House still ruled. His purpose was forcefully opposed by Rhudaur, which was at that time ruled by Hill-men who (as was only later discovered) had forged an alliance with the Witch-king's realm of Angmar.3
Faced with this threat from the east, Argeleb built fortifications on his eastern border, along the Weather Hills northward from the ancient Tower of Amon Sûl.4 The confrontation soon broke out into open war, the first direct conflict between the Dúnedain and Angmar, and in that war Argeleb himself was slain just seven years after becoming King. He was succeeded by his son, Arveleg I, who drove the enemy back from the Weather Hills and saved the realm, at least for a time.
The date of Argeleb's birth appears only in The History of Middle-earth volume XII, The Peoples of Middle-earth. It cannot therefore be considered completely reliable.
The ar- of Argeleb's name, as with his descendants, indicated his royal descent and claim to Kingship over all the lost realm of Arnor. The -geleb element is less clear, but it seems to be derived from celeb, meaning 'silver'. This possibility is reinforced by the fact that his name was originally drafted as 'Celebrindol', which definitely contains 'silver', until Tolkien chose to change it to 'Argeleb'.
We're not specifically told whether Cardolan supported Argeleb or not. At least they do not seem to have been directly opposed to the reunification of Arnor, and indeed they actively helped Argeleb's son Arveleg I to drive Arthedain's enemies back.
During their journey to Rivendell, Aragorn tells the Hobbits, 'The Men of the West did not live here [in the Weather Hills]; though in their latter days they defended the hills for a while against the evil that came out of Angmar.' (The Fellowship of the Ring I 11, A Knife in the Dark.) This is a direct reference to the fortifications made by Argeleb I nearly 1,700 years earlier.