Using Maverick's tags on Mountain Lion
Tags in Mavericks are actually partially backward compatible with labels in Mountain Lion, because they're implemented as sort of an extension (and renaming) of labels. The main issues are:


If you assign multiple tags to a file in Mavericks, Mountain Lion will only see the most recent tag (label)
Tag colors aren't preserved; e.g., if a red tag in Mavericks shares the same name as a blue label in Mountain Lion (say "Important"), it'll appear in blue on Mountain Lion


All that said, why can't you upgrade your Mountain Lion system to Mavericks? The system requirements didn't change, so any machine that can run Mountain Lion can be upgraded to Mavericks.