How can I determine the date of GIT cherry pick?

You can force both the author date and the commit date of a cherry-picked commit using something like

% GIT_AUTHOR_DATE='<desired author date>' GIT_COMMITTER_DATE='<desired committer date>' git cherry-pick <your commit hash>

This answer details the formats that you can use for the two date parameters.

A git commit has two dates. One is the author date and the other is the commit date. The default display is the author date. You want the commit date. git log -1 --pretty=fuller will show that or you can use %cD in a log format.