Paul is saying that he could endure any trial, perform any duty, cast out any wrong thinking, and conquer every temptation from the enemy. He was saying that whatever challenges came his way, he would be able to perform because of Christ in him. I believe that was not an arrogant statement, but a statement of faith. He relied confidently on Christ's strength, not his own strength.
I love the fact that as believers in Christ, under any trial, there is One who can strengthen us at every step of life. Christ is able to bring us victoriously through the trials of life. We have a Mighty Friend who is unchanging who will always help us.
We should not shrink from our duties or fear life, because in all circumstances, Jesus will uphold us.
How has Christ strengthened you and what areas in your life are you believing for Him to strengthen for the future? Ask and trust Him.