Is there a medication that can cure urticaria or anaphylaxis?

Depends on cause: A/u due to bee, hornet, wasp, fire ant sting can be cured with allergy shots. The patient gets the venom causing a/u in a series of injections. When the allergen can be identified - drug, food, pet, plant - complete avoidance "cures" a/u but accidental exposure results in more symptoms. In 50% of anaphylaxis & 90% of chronic urticaria the cause if never identified so cure is not yet possible.

Antihistamines: Usually help with the symptoms of itch and reducing the appearance of the rash.

Not really: There is no "cure" for urticaria or anaphylaxis other than to avoid the allergens that cause them, if they have or can be identified. Antihistamines are typically effective for treating urticaria, but true anaphylaxis is a medical emergency that is best treated by activating the emergency medical system.

No: Unfortunately, nothing makes them go away forever. If urticaria or anaphylaxis is due to food allergy, sometimes a patient can naturally "outgrow" the allergy. There are medical research on-going to see if we can actively "desensitize" patients to food allergen. But it is still in research state.

Hives: Although very difficult to cure hives and anaphylaxis especially if the cause is unknown, there are many treatment recommendations for controlling the hives such as high dose antihistamines and having an dpi pen available.