How to counter Karthus' ultimate?

The only complete item-based negation of Karthus' ultimate ability comes from Banshee's Veil and Zhonya's Hourglass. Even Hexdrinker will only mitigate (though, early game, the difference is moot) the damage dealt.

You can successfully mitigate the effects of Karthus' ultimate with items, runes, and masteries geared toward magic resist, however, since it's exclusively magic damage.

Champion abilities such as Nocturne or Sivir's spell shields can negate the effect. Morgana's Black Shield will mitigate the effect. Additionally, Soraka's ultimate, Wish, can mitigate the impact or save an impending death. A properly timed Hallucinate from Shaco (his ultimate), Elise (her Rappel) or Fizz's Playful Trickster can dodge the effects. Essentially, anything that makes you untargetable, when timed correctly, can negate the damage.

A properly timed Waltz of Blades (Fiora's ultimate) or Yi's alpha strike could potentially do it, but timing it would be very tricky. Someone should confirm those last two, however.

Quick Silver Sash will not stop Karthus' ultimate from hitting you. There used to be a glitch where it would remove the red animation, but it wouldn't actually prevent the ability from landing.

The best way is Silence/Taunt/knock back/Kill Karthus, before he finish his ultimate. But if that is not possible, here a list of items and champions that have the anti-karthus mojo.

EQUIPS

Nullify

• Zhonya's Hourglass: Places your champion into stasis for 2 seconds, rendering you invulnerable and untargetable but unable to take any actions.

• Banshee's Veil: Blocks one negative spell every 45 seconds.

Damage Reduction

• Hexdrinker: If you would take magic damage which would leave you at less than 30% of your maximum health, You gain a shield which absorbs up to 250 magic damage

• Maw of Malmortius: If you would take magic damage which would leave you at less than 30% of your maximum health, you first gain a shield which absorbs up to 400 magic damage

• Magic Resistence: Incoming magic damage is multiplied by a factor based on the unit's magic resistance:

SKILLS

Nullify

• Shaco's Hallucinate (Active): Shaco stealths for half a second and creates a clone of himself next to him... (When used in the right moment can avoid it)
• Sivir's Spell Shield (Active): Creates a magical barrier that lasts up to 3 seconds which blocks a single enemy ability used on Sivir. If an ability is blocked by the shield, Sivir regains 150 mana.
• Nocturne's Shroud of Darkness (Active): Nocturne creates a magical shield for 2 seconds which can block the next enemy spell that hits him.
• Kayle's Intervention (Active): Kayle bathes a target ally champion (or herself) in holy light, rendering them immune to all damage for a few seconds.

Damage Reduction

• Akali's Twilight Shroud (Active): Akali throws down a circular cover of smoke that lasts for 8 seconds. While inside the area, Akali gains armor and magic resistance
• Janna's Eye Of The Storm (Active): Janna conjures a defensive gale that shields her target from incoming damage for up to 5 seconds.
• Morgana's Black Shield (Active): Places a shield around target friendly champion, absorbing magic damage and preventing disables while the shield holds.
• Blitz's Mana Barrier (Innate): When Blitzcrank's life is brought below 20% health he activates Mana Barrier, creating a damage shield that lasts 10 seconds. The shield strength is equal to 50% of Blitzcrank's current mana
• Lux's Prismatic Barrier (Active): Lux shields herself for up to 3 seconds and throws out her wand to a target location
• Malphite's Granite Shield Malphite's skin innately produces a layer of rock as a shield which absorbs damage of up to 10% of his maximum health.

In early game game you can also get a red potion(elixer of fortitude,which gives you 140-235 health and 10 AD), and if you're low and karthus ults you just activate it and the damage of the ult won't do that much early game so you will survive it probably.