Important Note so that there is no confusion:
Christ is the Eternal God who is the creator (John 1:1 KJV), uncreated, and second person of the Godhead or Trinity. He is the eternal Word who was made flesh (John 1:14 KJV). For the Lord our God is one God who exists in three distinct persons (i.e. the Father, the Word, and the Holy Spirit) (1 John 5:7 KJV). Jesus Christ is God Almighty in the flesh and He is not a created angel as we would understand that word today. In the KJV: the word "angel" in reference to the title "Angel of the Lord" is actually translated as "Messenger" in the Hebrew. So the words as we would understand it today would say "Messenger of the Lord". So, no. Jesus is not an angel in our modern understanding. However, the Angel of the Lord is a title or a name. Anyways, I believe this "Messenger of the Lord" is a pre-incarnate appearance of Jesus Christ (Who is God) in the Old Testament. Again, these pre-incarnate appearances are called "Theophanies" or "Christophanies".
Jesus (Who is the Eternal God) is the Messenger of the Lord in the OT:
It is quite astonishing to discover that Jesus Christ appears not only in the prophecies and typifications within the Old Testament but He also actually makes real pre-incarnate physical appearances in the Old Testament, as well. Many of these appearances are under the title of the "Angel of the Lord" in the KJV. Yet, we clearly know that Jesus is not a created angel or being but He is God Almighty Himself (or the second person in the Godhead or Trinity).
Anyways, please take the time to look up the origin of the word "angel" at etymonline. While you are there, you will discover that it can also be defined as "messenger".
Online Etymology Dictionary
In other words, Jesus is the Messenger of the Lord in the Old Testament!
Now, where does our Lord Jesus appear as a messenger in the Old Testament?
1. The Story of Hagar -
Genesis 16:7-13
The angel of the LORD promised to do something that only God can do (see v.10). Hagar knew that it was the Lord who spoke to her (v. 13) and she identified the angel of the LORD as God: "Thou God seest me" (v.13).
Genesis 21:17-20
The angel of the Lord promised to do something only God can do (see v.18). This angel is identified as God (v.19).
2. The Story of Abraham & Sarah -
Genesis 18:1-33
Although the title of the "angel of the Lord" is not used here, it is clear within this chapter that is Jesus Christ. First, "the Lord appeared unto him (i.e. Abraham)" (v. 1) followed by the immediate appearance of three men (v. 2). Abraham and Sarah both call him Lord, as well (v. 3) (v. 12). The Lord also knew Sarah had laughed at God's promise when she was not present (v. 15).
3. The Story of Abraham & Isaac -
Genesis 22:15-18
The angel of the Lord is speaking (verses 15-16) and yet verse 16 makes it clear that it is God who is speaking ("saith the LORD")! In verses 17-18 the angel of the Lord promises to do what only God can do.
4. The Story of Jacob -
Genesis 31:11-13
The angel of God (v.11) identifies Himself as God: "I am the God of Bethel" (v.13).
Genesis 32:24-32
Jacob wrestles with a man who he later identifies the place with the name "Peniel", which means he had seen God face to face (v. 30).
5. The Story of Joseph -
Genesis 48:14-16
When Jacob was blessing Joseph and his sons he mentions :The Angel which redeemed me from all evil" (v. 16). Now, last time I checked, but angels do not redeem anyone. They are simply guardians.
6. The Story of Moses -
Exodus 3:2-7
The angel of the Lord appeared to Moses out of the midst of a burning bush and identified Himself as God (verses 4 and 6) and as Lord (verses 4 and 7). See also Acts 7:30-32 where the angel of the Lord is identified as the Lord God.
Exodus 14:19-21
The angel of God does what only God can do (verses 19-20) and is identified with the glorious manifestation of God in the pillar of the cloud (verses 19-20). In verse 21 this angel is identified as "the LORD".
Exodus 23:20-23
This passage makes it clear that the Lord’s angel is much more than a mere angel; this angel is closely identified with God: Lord’s "name is in Him" (v.21, "name" referring to Lord's nature and character) and God’s people must "obey His voice" (v.21). Indeed, He has the authority to "pardon your transgressions" or not to pardon them, and who can forgive sins but God alone? Notice that the angel is distinct from God and sent by God. The LORD said, "Behold, I send an angel before thee." We are reminded of New Testament parallels as the Son was distinct from the Father and sent by the Father (John 3:17; etc.) and yet equal to the Father (John 5:18) (John 10:30).
7. The Story of Balaam -
Numbers 22:20-35
Now in the story of Balaam, we begin with God speaking directly to Balaam, then a transition from God to "angel of the Lord" So guess who was standing before Balaam? None other than Jesus Christ. The "angel of the LORD" is used many times in verses 23-26, and verses 31-35 and the LORD in verses 28-31. Then continuing in Chapter 23 God meets Balaam in verse 4 and the LORD is mentioned in verses 5 and 16. These titles are being used interchangeably.
8. The Story of Joshua -
Joshua 5:13-15
Again, although the "Angel of the Lord" is not mentioned here, Jesus did appear to Joshua in the form of a man. This man was worshiped by Joshua (v. 14) and this man declared that Joshua was standing on Holy ground (v. 15).
9. The Beginning Story of the Judges -
Judges 2:1-3
The angel of Lord says things that only God could say. God is the One who brought them into the land which He swore to give unto their fathers (v.1). God is the One who promised to never break His covenant (v.1).
10. The Story of Gideon -
Judges 6:11-24
As you follow this passage, pay close attention to who is speaking to Gideon: in verse 12 the angel of the Lord is speaking; in verse 14 the Lord is speaking; in verse 16 the Lord is speaking; in verse 20 the angel of God is speaking. Lord and the angel of Lord are one and the same.
11. The Story of Samson's Birth -
Judges 13:3-23
Note especially verses 17-18. Manoah said to the angel of the Lord, "What is Thy Name?" (v.17) and the angel of the Lord said, "Why asketh thou thus after My Name, seeing it is secret?" The word "secret" may be translated "wonderful." It is the same Hebrew word found in Isaiah 9:6 – "His Name shall be called wonderful." In Isaiah 9:6 the term is used as a name of Christ who is also called "the Mighty God." The fact that the Lord's angel was God was certainly known by Manoah. After the angel of the Lord appeared to him Manoah said, "We have seen God!" (v.22).
Conclusion:
The Angel of the Lord is also mentioned in the story of David, Isaiah, the Psalms, and Zechariah. So it is amazing to see how Jesus Christ being our eternal God was always present in the past, even revealing Himself within the sacred Scriptures of the Old Testament, too.