There are different types if immortality. The simplest is not aging. Then there's immunity from disease, then invulnerability or near invulnerability. After that comes actual godhood, where you absolutely can never die.
Lately I've been wondering if alicorns really are immortal. Cadence seems to be aging normaly. Luna seems less mature than Celestia. If their ages were 10,000 and 9,993 you wouldn't think there would be a difference.
Maybe after the Nightmare Moon incident Luna was trapped in the moon not aging. And maybe Celestia found a way to magically extend her life so she could see her again.