The son of Isilmo, Tar-Minastir inherited the Sceptre of Númenor on the death of his childless aunt, QueenTar-Telperiën. His name Minastir means 'watchtower', and comes from the great westward-facing tower he built on the hill of Oromet near Andúnië, from which he would often look out towards the West. In Tar-Minastir's time the Elves were still beloved by the Númenóreans and welcomed by them, and Tar-Minastir had a close alliance with KingGil-galad of Lindon. However, it is also said that he envied their freedom from death, and from that envy a shadow of evil slowly grew over Númenor in the centuries following his rule.
The dating of Tar-Minastir's attack on Sauron is a little confused. We have good evidence that he sent out his fleet in II 1695, and that the Númenóreans defeated Sauron in II 1700. However, it's also well established that Tar-Minastir didn't inherit the Sceptre until II 1731, thirty-six years after he dispatched his fleet to Middle-earth.
It is difficult to reconcile these dates. We might conjecture that Minastir was acting as a regent or general for QueenTar-Telperiën, commanding Númenor's armies in her place. That's not impossible, but it's apparently contradicted by comments that explicitly make Tar-Minastir King at that time (for instance 'Then Tar-Minastir the King sent out a great navy...', from The History of Galadriel and Celeborn in Unfinished Tales). To make the timelines fit, we'd have to assume that 'Tar-Minastir the King' actually means 'Tar-Minastir, who later became King'. That's an awkward accommodation, but it seems necessary in order for Tar-Minastir's dates to make sense.