No, Psalm 119 is NOT the prayer of Jesus Christ the Son of God to His Father.
However, Jesus Christ is in Psalm 119….
He is the Word, the Commandments, the Statutes, the Precepts, the Law, the Testimonies and the Judgments of God that the psalmist is delighting in.
Psalm 119 is a type of prayer that someone like Joseph or Daniel would have prayed while in captivity. It’s what a Believer would pray in the Day of Trouble, Trials and Tribulations.
If you’re looking for a Psalm that Jesus Christ prayed to His Father, I’d look at…
Psalm 16, especially verse 10:
For thou wilt not leave my soul in hell; neither wilt thou suffer thine Holy One to see corruption.
https://www.biblegateway.com/passage/?search=Ps.16.10&version=KJV
You pointed out Psalm 16 which the NT writers reference as well especially the verse you mention. They also point out how it can not be David speaking since his grave was with them to that day when Peter was announcing the Gospel. This is a precedence for other Psalms as well. So although the Apostles and writers of the NT did not explain every Psalm, they left us examples on how we can understand them and Who is actually speaking in various Psalms.
Here is Psalm 16 with some headings to show it is the Son's prayer and who He is speaking to and about in the various verses.
Psalm 16
A Poem by David.
(Son to Father)
16:1 Preserve Me, 'El, for in You do I take refuge.
16:2 My soul, You have said to YHVH,
“You are My 'Adonay.
Apart from You I have no good thing.”
(Son about believers)
16:3 As for the holy who are in the earth,
they are the excellent ones in whom is all My delight.
(Son about wicked)
16:4 their sorrows shall be multiplied
who give gifts to another.
their drink offerings of blood I will not offer,
nor take their names on My lips.
(Son to and about Father)
16:5 YHVH assigned My portion and My cup.
You made My lot secure.
16:6 The lines have fallen to Me in pleasant places.
Yes, I have a good inheritance.
16:7 I will bless YHVH, Who has given Me counsel.
Yes, My heart instructs Me in the night seasons.
16:8 I have set YHVH always before Me.
Because He is at My right hand, I shall not be moved.
16:9 Therefore My heart is glad, and My tongue rejoices.
My body shall also dwell in safety.
16:10 For You will not leave My soul in Sheol,
neither will You allow Your faithful One
to see corruption.
16:11
You will show Me the path of life.
In Your presence is fullness of joy.
In Your right hand there are pleasures forevermore.
Psalm 16 is also the Psalm where He says His Father will show Him the path of Life. For Him that path of life was the Law and offering His life as a Sacrifice for our sins to save us. That helps explain v10 because even though He had to die in obedience to the Law to save us, His Father would not leave His soul in Sheol or to see corruption.
Yes, Psalm 16 is His prayer as NT writers point out and you mention here. It is also given to us as an example of the Son praying in other Psalms as well. Thanks to Peter and Paul and the writer of Hebrews for showing us these examples and of course the Holy Spirit for providing them so we can understand the Psalms.