Textbooks tell us that using the Simple Past in this scenario is more common in American English, and using the Present Perfect is more common in British English. Since the adverb of time "just" lets us know that the action happened recently, there's really no less information conveyed if you use the Simple Past. That said "one cup of it" isn't a natural-sounding collocation. "I (or I've) just had a cup" would sound much more natural..

Why do we do this to ourselves? 😂 Yes, both are used interchangeably. But, if I was to think about "rules" of English grammar I would use "I've just had" (Present Perfect) because the time of drinking the cup of tea is not specified or even important. Plus, it seems to have an influence on the present moment. I assume that the speaker is satisfied in the present moment because of finishing a cup of tea in the recent past. "I just had" (Simple Past) indicates a completed action that took place and was finished in the past. I just had a cup of tea at 7:00 a.m. Again, either response would be used without analysis. That's my two cents! But, I am open to any other ideas.

These sound natural: "I've already had a cup." "I just had a cup." For some reason though, "I just have had a cup" sounds less natural. "Just" changes things a bit because it somehow turns your cup into a historical event.

Either way is colloquial and easily understood.