No, men were not ordered to sink with their ships. The figure you're looking at is mostly men who were killed during the attack on the ship or who died in the aftermath, fighting fires or trying to escape. Most of Akagi's crew was saved. It's likely that some men drowned or died in accidents while trying to abandon ship, that some wounded men were unable to leave the ship or be saved, that some may have been trapped by damage, debris, or fire, and even that some small numbers of men chose to go down with the ship. But the large majority of Akagi's crew was taken off by other Japanese ships. She was actually more fortunate than the other Japanese aircraft carriers at Midway; she had suffered only one direct hit from an American bomb.

The captain of Akagi, Taijiro Aoki, attempted to go down with his ship but was eventually discouraged or ordered not to do so. Captain Ryusaku Yanagimoto of Soryu chose to accompany his ship to the bottom. Rear Admiral Tamon Yamaguchi and Captain Tomeo Kaku, both aboard Hiryu, delivered a speech to their men on the flight deck and ordered them to leave the ship but chose not to leave themselves. The captain of Kaga, Jisaku Okada, was killed during the attack.

Admiral Chuichi Nagumo, commander of the strike force, and his chief of staff, Rear Admiral Ryūnosuke Kusaka, both evacuated Akagi and eventually transferred command to the cruiser Nagara. Other crew would have found their way into destroyers, which were easier to maneuver into position than cruisers or other larger ships.

Anthony Tully's website has a robust description of Kaga's activities after being damaged. The cruiser Chikuma sent one of its launches (a small boat carried aboard the ship) to help rescue men, and destroyers Hagikaze and Maikaze also stood by to collect men. There's also some good detail on Soryu's end; destroyers Hagikaze and Isokaze both took survivors off before firing three torpedoes into her to sink the ship.

Many of Hiryu's men were rescued in a relatively orderly evacuation into the waiting destroyer Makigumo and Kazigumo. Tully's site reports that there were about 70 men still aboard Hiryu when a Japanese plane saw it the following morning. This wouldn't have meant they were ordered to stay aboard, though -- the Japanese sent a ship to rescue them, but it sank before the ship arrived (some were later pulled out of the water by American ships).

The process of abandoning ship would also include destroying some items value, though that would depend on the situation. A code book might be burned or placed in a bag filled with weight and tossed overboard so there was no chance of recovering it, for example. Once the crew was off and the attending destroyers had moved to a safe distance, they would fire torpedoes at the ships to sink them. This didn't always go according to plan; sometimes torpedoes malfunctioned or ships stayed withstood more damage than expected, but eventually they would sink.

Tully's book on Midway, "Shattered Sword" is worth reading if you're interested in the battle.