Show Perspective
It was just something a warrior would say prior to the battle.
He said:
Now it begins
He meant that now their fight begins.
Ned replied:
No, Now it ends
Ned was here stating the obvious that this was the last battle of Robert's Rebellion and after it, the long war would end.
There is no special hidden meaning to what Arthur Dayne said.
And not all last words are meaningful. Remember Tyrion's "Last words" when he thought executioners had come for him at last?
As the keys rattled and the door to his cell pushed inward, creaking,
Tyrion pressed back against the dampness of the wall, wishing for a
weapon. I can still bite and kick. I'll die with the taste of blood in
my mouth, that's something. He wished he'd been able to think of
some rousing last words. "Bugger you all" was not like to earn him
much of a place in the histories.
A Storm of Swords -
Tyrion XI
Or the mostly absurd/meaningless last words here in this complete list of last words by GoT actors?
When he wished Ned luck in the wars to come, there are three possible reasons for it:
Arthur Dayne was known as personification of chivalry. It was just a chivalrous gesture and sits well with his character.
Arthur Dayne subtly implied that the war was far from over and eventually the exiled Prince Viserys and his unborn sibling (As Daenerys was not born then) would come back to retake their right.
Arthur obviously knew that:
The child in the tower, Jon, was Rhaegar's son and he was implying
that war will continue when the boy or his uncle/aunt make their claim
Books Perspective
Not to mention, he never said that in the books. He was courteous, but he didn't say that "I wish you luck" thingie as it is, as you observed, tantamount to giving up.
They however must have known that they were going to be killed as they had decided that they were not going to flee. It was about Honor of Kingsguard and a Kingsguard doesn't flee and gives his life for those of the royal blood.
Full conversation:
“I looked for you on the Trident,” Ned said to them.
“We were not there,” Ser Gerold answered. “Woe to the Usurper if we
had been,” said Ser Oswell.
“When King's Landing fell, Ser Jaime slew your king with a golden
sword, and I wondered where you were.”
“Far away,” Ser Gerold said, “or Aerys would yet sit the Iron Throne,
and our false brother would burn in seven hells.”
“I came down on Storm's End to lift the siege,” Ned told them, and the
Lords Tyrell and Redwyne dipped their banners, and all their knights
bent the knee to pledge us fealty. I was certain you would be among
them.”
“Our knees do not bend easily,” said Ser Arthur Dayne.
“Ser Willem Darry is fled to Dragonstone, with your queen and Prince
Viserys. I thought you might have sailed with him.”
“Ser Willem is a good man and true,” said Ser Oswell.
“But not of the Kingsguard,” Ser Gerold pointed out. “The Kingsguard
does not flee.”
“Then or now,” said Ser Arthur. He donned his helm.
“We swore a vow,” explained old Ser Gerold.
Ned’s wraiths moved up beside him, with shadow swords in hand. They
were seven against three.
“And now it begins,” said Ser Arthur Dayne, the Sword of the Morning. He unsheathed Dawn and held it with both hands. The blade was
pale as milkglass, alive with light.
“No,” Ned said with sadness in his voice. “Now it ends.”
A Game of Thrones - Eddard X
