The time between Samuel's first meeting with Saul and his anointing appears to have been a matter of about one day.
To morrow about this time I will send thee a man out of the land of
Benjamin, and thou shalt anoint him to be captain over my people
Israel, that he may save my people out of the hand of the Philistines:
for I have looked upon my people, because their cry is come unto me.
(1 Samuel 9:16, KJV)
And Samuel answered Saul, and said, I am the seer: go up before me
unto the high place; for ye shall eat with me to day, and to morrow I
will let thee go, and will tell thee all that is in thine heart. (1
Samuel 9:19, KJV)
Then Samuel took a vial of oil, and poured it upon his head, and
kissed him, and said, Is it not because the LORD hath anointed thee to
be captain over his inheritance? (1 Samuel 10:1, KJV)
From there to his coronation may have been nearer to a year, because there was an initial delay when the people who had first asked for a king became upset at the selection having come from the tribe of Benjamin, and not being one from among themselves. During this time, Saul, already chosen to be king, meekly returned home to his farm. Not long afterward, it was a military crisis that brought him into action, and all Israel acknowledged him as their king.
How Old Was Saul at His Coronation?
Because the Bible does not tell us this, we can only attempt to estimate his age based on the facts of the story.
According to Paul, in the book of Acts, King Saul reigned 40 years.
And afterward they desired a king: and God gave unto them Saul the son
of Cis, a man of the tribe of Benjamin, by the space of forty
years. (Acts 13:21, KJV)
Saul's son Ishbosheth began reigning when Saul died, and the record says he was 40 years old.
Ishbosheth Saul's son was forty years old when he began to reign over
Israel, and reigned two years. But the house of Judah followed David.
(2 Samuel 2:10, KJV)
So Ishbosheth would have been born during Saul's first year as king.
We aren't given much to go on regarding Saul's age at any point in his life, but we know he was above marriageable age when he became king, for he was at the point of fatherhood.
But Ishbosheth was not Saul's firstborn.
Now the sons of Saul were Jonathan, and Ishui, and Melchishua: and the
names of his two daughters were these; the name of the firstborn
Merab, and the name of the younger Michal: (1 Samuel 14:49, KJV)
Whether the daughter Merab was born before Jonathan is unclear, but that Jonathan was the eldest son, being listed first among Saul's sons, seems plain.
Another record of Saul's sons also lists Jonathan first.
And Ner begat Kish, and Kish begat Saul, and Saul begat Jonathan, and
Malchishua, and Abinadab, and Eshbaal. (1 Chronicles 8:33, KJV)
Nor are these the only sons, as we see the record of sons slain with Saul in that final battle.
And the Philistines followed hard upon Saul and upon his sons; and the
Philistines slew Jonathan, and Abinadab, and Melchishua, Saul's sons.
(1 Samuel 31:2, KJV)
After this, Ishbosheth, Saul's son, is brought to the throne--not having been slain with the other sons of Saul.
Jonathan himself is a bit of an enigma. He was already old enough by Saul's second year as king to command a battalion of 1000 soldiers.
Saul reigned one year; and when he had reigned two years over Israel,
(1 Samuel 13:1, KJV)
Saul chose him three thousand men of Israel; whereof two thousand were
with Saul in Michmash and in mount Bethel, and a thousand were with
Jonathan in Gibeah of Benjamin: and the rest of the people he sent
every man to his tent. (1 Samuel 13:2, KJV)
And Jonathan smote the garrison of the Philistines that was in Geba,
and the Philistines heard of it. And Saul blew the trumpet throughout
all the land, saying, Let the Hebrews hear. (1 Samuel 13:3, KJV)
This means that Saul must have been close to 40 years old or more in order to have had a 20+ year-old son. But Jonathan also had a 5-year-old son (Mephibosheth) at the end of Saul's reign, nearly 40 years later.
This is recorded at the death of Saul and Jonathan:
And Jonathan, Saul's son, had a son that was lame of his feet. He was
five years old when the tidings came of Saul and Jonathan out of
Jezreel, and his nurse took him up, and fled: and it came to pass, as
she made haste to flee, that he fell, and became lame. And his name
was Mephibosheth. (2 Samuel 4:4, KJV)
Jonathan may not have married young, or it is possible that he was not fully of marriageable age when he became a soldier. Being a soldier from the beginning of Saul's reign, however, should indicate that Saul himself, Jonathan's father, must have been nearly 40 years old at his coronation.
If Saul had been only 30, as some suggest, when he became king, then one must assume either that he was a father when but a teenager, or that his son was a military captain as a pre-teen. Had Saul been 16 upon becoming a father, Jonathan would have been around 14 to 16 as a captain. These ages are unreasonably young.
Assuming Saul had met the prophet Samuel only a short time prior to his anointing (within the same year), and based on the military activities of Saul's son Jonathan within the first two years of Saul's reign, we can infer that Saul himself must have been nearly 40 years of age, or greater, when he met Samuel and became king.