can I upgrade Mountain Lion to a Mac 10.5.8

It depends upon your Mac. Get more information about your computer. Go to the Apple in the upper left corner of any window, then "About this Mac" and "More Info..." Copy and paste the information here, but omit the serial number and Hardware UUID (if present).

Upgrading to Mountain Lion - http://www.apple.com/osx/how-to-upgrade

Mac OS X: System requirements for Mountain Lion (10.8) - http://www.apple.com/macosx/specs.html

Lion and Mountain Lion application compatibility - http://roaringapps.com/apps:table

Kappy 08/2012 post on upgrading to Snow Leopard, then Lion or Mountain Lion - https://discussions.apple.com/thread/4246342?answerId=19401628022#19401628022 - including how to get Snow Leopard and Lion since Apple removed them from the online store.

Incidentally, you posted this in the 10.3 and earlier forum. Nobody is going to see it here so it would be good if you selected the correct forum next time you ask a question. 🙂

If the Mac is compatible, it needs to be upgraded through 10.6(phone the online Apple Store) and then updated to 10.6.8 first.

(70521)