The age of Solomon when he began to reign is unknown. However, we can work out some dates and establish an approximate age. The following dates are explained in The Times (chapter 3)
|1011BC||David starts to reign, age 30 (2 Samuel 5:4-5)|
|971BC||David dies, age 70 (1 Kings 2:11)|
|971BC||Solomon starts to reign (1 Kings 2:12)|
|931BC||Solomon dies (1 Kings 11:42)|
Solomon was born well after David became king, although we can’t be sure of the date. We do know that it was probably several years after David moved to Jerusalem: he appears to have been in Jerusalem for some time when he met Solomon’s mother Bathsheba (2 Samuel 11:1-3), and then there has to be another couple of years to allow for Bathsheba’s first pregnancy, marriage to David, and then a second pregnancy. So that places Solomon’s birth at least 10 years into David’s reign, which means that Solomon was under 30 when he became king. Shortly after he became king he married the daughter of Pharaoh (1 Kings 3:1) which would suggest he was probably at least 20 years old.
He describes himself as a “little child” (1 Kings 3:7), although that is surely an exaggeration.
In summary, we can be reasonably confident that Solomon was in his 20s when he became king, but it is not possible to be more precise.