Mr Ban is expected to hold talks with Israeli and Palestinian leaders.
In the latest in a wave of attacks, an Israeli soldier was stabbed and wounded and his Palestinian assailant shot dead in the occupied West Bank, Israel said.
Eight Israelis and more than 40 Palestinians, including attackers, have been killed in recent weeks.
The upsurge in violence began last month when tensions at a flashpoint holy site in Jerusalem revered by Jews and Muslims boiled over amid rumours that Israel planned to relax long-standing rules to strengthen Jewish rights at the complex. Israel has denied such claims.
‘Weapons of despair’
Ahead of his surprise visit to the region on Tuesday, Mr Ban urged both Israelis and Palestinians to end their “posturing and brinkmanship” and return to trying to achieve a two-state solution to the conflict.
Addressing Palestinian youth, he said he understood their frustration and anger “at the continued [Israeli] occupation and expansion of settlements”. But he added: “You must put down the weapons of despair.”
Mr Ban told Israelis that he understood their “genuine concern about peace and security”, but warned that “walls, checkpoints, harsh responses by the security forces and house demolitions cannot sustain the peace and safety that you need and must have”.
On Tuesday morning, an Israeli soldier was stabbed and lightly wounded during clashes with Palestinians near Beit Awwa, a village west of Hebron, the Israeli military said.
The Palestinian attacker was shot and killed, according to Israeli medics.
Later, an Israeli man died of his injuries after being run over at the al-Fawar junction, south of Hebron, Israeli medics said.
An Associated Press photographer who witnessed the incident said the man left his car after Palestinian protesters threw stones at it.
He then began to hit passing Palestinian cars with a large stick similar to a baseball bat, the photographer said. The man hit a lorry with the stick, and the lorry ran the man over, he added.
A Palestinian lorry driver later handed himself in to Palestinian police in nearby Dhahiriya, saying he hit the man by accident, according to the Palestinian news agency, Maan.
Overnight, Israeli troops destroyed and sealed the home of a Palestinian man who ran down, stabbed and killed an Israeli woman in the West Bank last year.
The Israel Defense Force (IDF) said the demolition of the third-floor flat in Hebron where Maher Hashlamoun’s family lived was a “clear and deterring message that the involvement in terrorism has a price”.
Israel has said it will speed up carrying out demolition orders on the homes of Palestinians who have carried out deadly attacks.
The practice has been denounced by human rights groups as a form of collective punishment and against international law.