Do any weapons do extra damage against the final boss?

While one could argue that Dragonrend is "super-effective" against Alduin, the most obvious choice is the unique Blades sword "Dragonbane". Dragonbane can be found in Sky Haven Temple, located on a table off to the left of the main chamber along with a full set of Blades armor and several Blades sword.

Unfortunately, Dragonbane's enchant is unique so you won't be able to enchant it yourself. Like most unique items, Dragonbane is leveled and the power of the enchantment is determined when you encounter it. It can range from 20-40 extra damage against dragons.

Note that Alduin will have a special voice response if you carry Dragonbane in your inventory into your battle with him. Dragonbane merely has to be present in your inventory, not equipped.

The only weapon that do extra damage to a dragon is a unique 2-handed sword called "Dragonbane" which you can find in Sky Haven Temple. It can do about 30-40 extra damage against a dragon.

But can you kill Alduin easily with Dragonbane? It all depends on your character's skills level and specialty. IMO a well enchanted weapon is always a better choice than Dragonbane.

PS : You can avoid taking damage from dragon breath attacks easily by using Spellbreaker or magic wards. Also try to avoid from taking direct hit from Alduin's fire rain-balls.