The sentence refers to a point in the future (next year), and the structure "By this time next year" indicates a future perfect tense. The future perfect tense is used to describe an action that will be completed at some point in the future. "Will have been" is the correct form here because it expresses that, by a certain future time, the action of turning twenty years old will have been completed.

So the sentence would be: By this time next year, I will have been twenty years old.