Q. In John 1:47-51, why did Jesus use the phrase "without guile" to describe Nathaniel’s character? And did Jesus physically see Nathaniel under the fig tree or did He see him while praying prior to their actual meeting? Finally, why did Nathaniel, after what seems to be only a brief exchange of greetings, acknowledge Jesus as the Son of God? I wouldn't expect Christ's words would be enough to lead Nathaniel to immediately acknowledging Jesus as the Son of God.
A. In John 1, Jesus meets Nathaniel. Jesus declares as He meets Nathaniel that Nathaniel is an Israelite in whom there is no deceit (or guile). This statement had double meaning and was a play on words. First, Nathaniel had earlier commented to Philip that the Messiah couldn't be from Nazareth, because "can anything good come out of Nazareth?" Obviously, Nathaniel's statement demonstrated a complete lack of faith in Jesus.
In response to Nathaniel's cutting comment concerning Jesus, the Lord offered His own cutting assessment of Nathaniel. Jesus said that Nathaniel was an Israelite in whom there was no deceit, but the use of the Greek word dolos (which means deceit or crafty) was a play on the name of Jacob, whose name meant deceiver or crafty in Hebrew.
Jacob became Israel, the father of the twelve tribes of Israel, so what Jesus said, in effect, was Nathaniel was an Israelite in whom there was no "Jacob" or no true Israel. He was an Israelite by birth, but by his lack of faith Nathaniel was not a true descendent of Israel spiritually.
Secondly, Jesus was speaking prophetically about what Nathaniel would speak next. After Jesus confronted him and revealed Himself to be the Christ, Nathaniel declared that Jesus was the Son of God, the King of Israel. By this statement, Nathaniel spoke truly, and therefore Nathaniel's declaration was trustworthy and in him was no deceit.
To your second question about Jesus' vision, it's clear that Jesus "saw" Nathaniel in a spiritual sense prior to ever meeting him. Notice in verse 47 that Jesus noticed Nathaniel coming to Jesus, which means that Nathaniel had already left the tree and was walking to meet Jesus. Jesus makes His cutting remark about Nathaniel as He watches Nathaniel approaching Him, and then as they talk Jesus confides that He knew of Nathaniel from long ago.
Jesus is referring to God's election of His children before even the foundation of the Earth. Paul describes this knowledge in Ephesians 1:
Eph. 1:3 Blessed be the God and Father of our Lord Jesus Christ, who has blessed us with every spiritual blessing in the heavenly places in Christ,
Eph. 1:4 just as He chose us in Him before the foundation of the world, that we would be holy and blameless before Him. In love
Eph. 1:5 He predestined us to adoption as sons through Jesus Christ to Himself, according to the kind intention of His will,
Eph. 1:6 to the praise of the glory of His grace, which He freely bestowed on us in the Beloved.
Jesus demonstrated His Deity to Nathaniel simply in revealing to Nathaniel something that no mere man could have known: that Philip had called to Nathaniel while Nathaniel was sitting under a fig tree. Nathaniel knew Jesus had not been present in that moment nor had Philip left Nathaniel's side since then, so how could Jesus have known this information? Jesus' statement shocked Nathaniel, and prompted him to declare that Jesus was the Son of God.
Was Jesus' simple statement the thing that converted Nathaniel? Not exactly. The Holy Spirit must have been at work in Nathaniel prior to the moment of Jesus arrival so that once Jesus appeared and spoke to Nathaniel, his response would be assured. This is the meaning of Paul's statement in 1Cor 12:3:
1Cor. 12:3 Therefore I make known to you that no one speaking by the Spirit of God says, “Jesus is accursed”; and no one can say, “Jesus is Lord,” except by the Holy Spirit.
All men are brought to a saving faith in Christ in precisely the same manner: we are drawn to Christ by the Holy Spirit, Who convicts all men of sin and brings a repentance that leads to salvation. This repentance is the work of God by His Spirit, and it prepares the heart to receive the Gospel (see Acts 11:18; Rom 2:4; 2Cor 7:9; 2Tim 2:25). Obviously, Nathaniel had received the call of the Spirit, so that once Jesus spoke to him, Nathaniel was ready to believe. It only required that Jesus reveal Himself to Nathaniel for his faith to be established.