I need to know about the information for the external user last logged-in time over a site. Is it possible to get through REST or Client-Object Modal?
Since you know what site they'll be using, you could turn on audit logging and check the last time they trigger a basic audit case like:
Opening or downloading documents, viewing items in lists, or viewing item properties
This of course assumes that when people log in they then open or view an item (probably a page...), but I think that is a fair assumption.
You need to have this setup beforehand though (won't be able to go back and find people from before you turn auditing on).
Turn on audit logging under Site Collection Administration > Configure Audit Settings
When you want to know info about someone generate an audit report for that specific user and use some good old fashioned excel skills to sort the sheet by event time.