When (day, month, year) did Adoniahu die, as according to the pasuk below?
From pshat, it happened shortly after Shlomo was crowned. According to Yalkut Me'am Lo'ez on Melachim, this was in the Hebrew year 2924. Similarly, the Yalkut says that Adoniyahu came to Bat-Sheva just a few days (ימים אחדים) after the passing of David. Me'am Loez also says that David died on the Atzeret (Shavuot), which came out on Shabbat that year.
From all of this, it seems that Adoniyahu was put to death sometime in Sivan of the year 2924.
If you're willing to entertain a more pilpulistic approach, then we can take the calculations further: Me'am Loez says that it was "ימים אחדים" after the death of David, and מיעוט רבים שניים - the smallest plurality is two, so it would have been exactly two days after Shavuot of that year. However, at the time, Shavuot, like every chag and month, was calculated according to the sighting of the new moon, so Shavuot didn't necessarily fall on the 6th of Sivan (see here). Despite not knowing on what day of Sivan exactly Shavuot fell on that year, I think we can still safely say that Adoniyahu's death occurred somewhere between the 7th and the 9th of Sivan of the year 2924.