Using old 2008 Mac as a second monitor

Sorry no, Target Display Mode only works with a few 27” and 21.5” model iMac’s.

for a list of compatible iMac’s refer to > Use your iMac as a display - Apple Support

No. Target display mode requires a 2009 to 2014 iMac.

See here: Use your iMac as a display - Apple Support

And you would not want to anyway, since the supported resolution on a 24" 2008 iMac is way lower than what the 2019 MacBook Air can output, so it would not produce the best results.