The number of even proper divisors of the number $1008.$ 

 $\because 1008=2^4\times 3^2\times 7^1.$

$\therefore$  Required number of even proper divisors 

    $=$ Total number of selections of atleast one $2$ and any number of $3\text{'}s$ or $7\text{'}s.$

      $=4\times (2+1)\times (1+1)-1=23$