Zenbu will let you filter entries by
edit_date (similar to the Edit page in ExpressionEngine, see screenshot below).
If you don't want to use an add-on you could run a MySQL query to check the
exp_channel_titles. Something like:
SELECT entry_id, title, DATE_FORMAT(edit_date, '%m-%d-%Y-%k:%i') AS updated
ORDER BY edit_date DESC LIMIT 50
This above just returns a list of entries along with the edit date, with the most recently edited entry listed first. You can easily run this query in the EE Control Panel using the Database Query Form, or with an application like Sequel Pro or Navicat.
Note that there will likely be other data besides the edit date entry data you'd want to check before you import a copy of the staging database (changes in categories, members, and comments for example). But if you know for sure that the only data changing on your staging site is entries then you should be good just checking