But this sentence is really unnatural. I can't think how one could modify a state like health or goodness with an adverb of time like already unless some change is indicated by a different verb or another adverb such as "again." To convey the meaning of this sentence we would have to say something like "is already healthy again" or "has recovered already."
"I'm feeling better" can be ambiguous without an extra qualifier - all better vs a little better or a lot better, all of which land at different spots on the progression of healing. But without any qualifier or tone of voice to add nuance, "I'm/he's/she's better" on it's own will generally be understood to mean full recovery.
In idiomatic English, certainly, it sure seems like this is the best translation, unless I'm missing something about the Russian. If not, does anyone have different Russian that would mean "My sister is already better"? I'm guessing "лучше" doesn't work for phrases like "get better soon" that clearly only refer to health!
You can use Моей сестре уже лучше. It means a different thing, though: we say someone is "лучше" both for people who are back to normal and for people who are still ill but are feeling better than previously.
Getting better is easily translated as выздоравливать or поправляться.
I, however, have my doubts whether a learner would easily come up with either of these two given the English phrase with "better" (we do not even teach them anywhere).