How to determine when an app was downloaded and installed
App information can be viewed using iExplorer:



Clicking on the disclosure triangle reveals different dates, and I assume the most recent date on a folder (excepting the tmp folder in this example) is the date of the last update, either when released or installed.

What's interesting from this screenshot is that Chrome was loaded in the background, but I was asleep and not using the iPhone when tmp was updated (2/24/15, 1:26 AM.)

You could also check the dates on the .ipa files in ~/Music/iTunes/MobileApplications, but I would assume the dates would be correct only if you updated your apps in iTunes and uploaded to the iPhone with USB/Wireless and not OTA on the phone.