In our text of 1 Sam 16:13 we find,
1. Samuel anointed David, rather than Jesus sending the Holy Spirit.
2.David was anointed with oil, the symbol, the type, of the Holy Spirit, on the day of Pentecost the Holy Spirit appeared like tongues of fire on all of the gathered believers
3. It was only David amongst his brethren that was anointed, it was not universal.
4. The Holy Spirit was not in David, it came upon him, on the day of Pentecost the believers had already received the Holy Spirit, John 20:22 He was in them and then the Holy Spirit came upon them.
Let's look again at the New Covenant,
Behold, the days come, saith the LORD, that I will make a new covenant with the house of Israel, and with the house of Judah: Not according to the covenant that I made with their fathers in the day that I took them by the hand to bring them out of the land of Egypt; which my covenant they brake, although I was an husband unto them, saith the LORD: Jer 31:31, 32
David was one of the fathers, the New Covenant was different from the Old Covenant of Sinai, the difference between the two covenants is the first was conditional on the people doing their part, the second is unconditional, it's God doing everything for us.
Here's something else to consider,
(But this spake he of the Spirit, which they that believe on him should receive: for the Holy Ghost was not yet given; because that Jesus was not yet glorified.) John 7:39
How could David 1000 years before Christ or any of the Old Testament Saints receive the Holy Spirit before the 17th of Nisan 32A.D.?
We read the Holy Spirit came upon David, Samson, Othniel, and others to empower them to fight their enemies, we read in the New Testament the Holy Spirit came upon believers to preach the Gospel, which points out a definitive point for us, the Old Testament is physical and material, the New Testament is spiritual and eternal.
I hope I have answered you question.
Blessings