The iProperties (in your screenshot) are set manually, so unfortunately, you won't see them change when saving.
There is a "Last modified" property on the drawing file which records the last time you saved it (see the "General" tab of iProperties). To access it though, you'll need a line of iLogic to push the date into either a custom iProp, or a standard iProp - whichever is your preference.
I use a slightly simpler approach and grab the 'now' variable, capturing only the date (and you can mess with the formatting of the date easily by swapping the text around):
iProperties.Value("Custom", "Last Modified Date") = Date.Now.ToString("dd-MM-yyyy")Set a trigger to run the rule "Before Save", and then save it into your drawing template.
You mention "check in" - are you using Vault? If so, Vault Professional has a whole Revision table function. If you're using any other version, then "Checked in" date would be a good property to search by, but not something you can write back to the drawing.
Peter