If the Update button isn’t showing on your Kindle, clear your cache and then force stop the Amazon Appstore in the Kindle's Settings.
This should cause the Appstore app to restart, and cause the update button to show up.
How to clear the cache:
Open the Kindle's settings
Select Apps & Games
Select Manage All Applications
If Running Applications or Third-Party Applications is selected, swipe until "ALL" applications are displayed
Scroll down and select Appstore
Tap on "CLEAR CACHE"
Tap on "FORCE STOP"