Minecraft wiki entry for Ultra_hardcore_survival, associated text: Enchant your sword, bow, and armor (all preferably diamond/netherite). Armor is especially important; go for Protection IV.
Players can spawn on an obsidian platform which floats above the void. This leaves players very vulnerable to being knocked into the void by the ender dragon. To reduce this risk, keep an ender pearl on your hotbar while entering the End, so that you can teleport off the platform. Remember though that when you do this you will also take ender pearl damage.
Throwing an ender pearl and then eating a golden apple afterward will negate all damage given, plus give 2 extra hearts.
Bringing in some potions can assist you in the fight:
Night vision makes it easier to spot the dragon in the dark
Slow falling negates fall damage when the dragon flings you in the air and gives you more time to place an ender pearl in case you get kicked off the island. Definitely use one if you don't feel safe using water buckets to break your fall!
Strength, speed and regeneration potions can come in useful too.
Always use arrows or other projectiles to destroy end crystals, rather than melee weapons. Two of the End Crystals will always be enclosed by iron bars, but this can be bypassed by getting close to their towers and shooting them from the opening at the bottom of the iron bars.
If you're prone to aggravating endermen, wear a carved pumpkin on your head. This will narrow your vision however, so be mindful of where you're walking. Pressing F1 while wearing the pumpkin as a helmet hides the pumpkin face, but will also hide your hotbar, health bar, and hunger bar. Alternative resource packs can make the pumpkin face texture transparent.