to illustrate the solution to this problem using both Statdisk and the table in your text
A 99.5% confidence interval is pictured as the area in the HUMP of the normal curve - what is left is the area in the two TAILS combined. Each TAIL has half of this remaining area, and that is where ALPHA/2 comes from.
How to use the table or Statdisk to find the z score for a 99.5% confidence interval -
Source: Triola