In 1 Corinthians 1, the term "Jesus Christ" appears 6 times while the term "Christ Jesus" appears 4 times. Is there any significance in the difference in the way Paul uses these two terms?
The more common nomenclature in the New Testament is actually 'Christ Jesus', something like 90 times against about 70 references to 'Jesus Christ'.
The contrast in the emphasis between the occasions when the title 'Christ' is first or second in the two names, can be seen in 2 Timothy verses 9 and 10, quoting from the KJV :
(v9) ...[God] ... his own purpose and grace, which was given us in Christ Jesus before the world began,
(v10)... But is now made manifest by the appearing of our Saviour Jesus Christ,
In verse 9, the emphasis is on the calling of God in Christ and the purposes of God in Christ before the world began, so the title (which is a matter of priesthood and anointing and headship) is stated first, before the personal name of Jesus.
In verse 10, the emphasis is on the coming into the world - the appearing - of the Saviour and here his name (Jehovah-Eshua, Jehovah-salvation) is placed first, thereafter his messianic title.
The distinction is a subtle one in different contexts regarding which ministrations of the Lord are in view : his Saviourhood, his Priestliness, his Royal rule over his own people, and his Headship over a new humanity.
Paul, later in the epistle uses the full, risen and ascended title in chapter 4 verse 1 :
I charge thee therefore before God, and the Lord Jesus Christ, who shall judge the quick and the dead at his appearing and his kingdom ... [KJV]
In this case, Paul is concerned with the coming of the Lord from his place in ascended glory to judge all humanity at the end of time and he gives the full title of both Lord and Christ in such a context.
This is the King of Kings and Lord of Lords, who shall, at the time the Father so appoints, cause all of the enemies of Jesus Christ to be beneath his feet, 1 Corinthians 15:25.