解:$(1)$正确,原因是一个数的倒数的倒数等于其自身;
$(2)$原式的倒数为$(\frac {1}{3}-\frac {1}{6}+\frac {3}{8})÷(-\frac {1}{24})$
$=(\frac 13-\frac 16+\frac 38)×(-24)$
$=-24×\frac {1}{3}+24×\frac {1}{6}-24×\frac {3}{8}$
$=-8+4-9$
$=-13$
则$(-\frac {1}{24})÷(\frac {1}{3}-\frac {1}{6}+\frac {3}{8})=-\frac {1}{13}$