"Hold on to" here means "firmly believe." Alternate translation: "Continue to firmly believe the word of life" (See: [[rc://en/ta/man/jit/figs-metaphor]])
The phrases "run in vain" and "labor in vain" here mean the same thing. Paul uses them together to emphasize how hard he has worked to help people believe in Christ. Alternate translation: "I did not work so hard for nothing" (See: [[rc://en/ta/man/jit/figs-parallelism]])
The scriptures often use the image of walking to represent conducting one's life. Running is living life intensely. (See: [[rc://en/ta/man/jit/figs-metaphor]])