Is it possible to update an existing Windows Phone 8 app to Windows Phone Store 8.1

TL;DR; - It preserves data on Isolated Storage when updating from WP8.0 to WP8.1 Runtime.

Because the provided link to MSDN says only about Silverlight apps, and it's not clear (if I hadn't missed something) what would happen in case: I've old WP8.0 Silverlight App and now I decided to upgrade it to WP8.1 Runtime - I've performed a simple test:

1. I've published a Beta version of the App - WP8.0 Silverlight.
2. After successful instalation on the Phone, I've copied some files inside the App, so that there is something on Isolated Storage
3. I've submitted an update - went to Store, selected Beta version (WP8.0), clicked Update then Upload and describe your packages, after a while, Add new (not replace the old WP8.0 package), add some information, save and submit.
4. After some time my Phone is notified that there is an update for the App - I click yeah - update
5. After successfull instalation, I see that it's a new App, I click my special button to see what is on Isolated Storage and I see that there are old files from WP8.0 version

So it turns out that LPains answer is correct (+1).

Do you mean update the code or update the application in the store?

If you mean update the code, you have to recreate it from scratch. You may be able to use some content but a lot has changed on the new Windows Phone 8.1 (Windows Runtime) including how you write Xaml.

If you meant update in the store than yes, that will work. When you update your app with a new one on the new format the store automatically understands and migrates the IsolatedStorage data: http://msdn.microsoft.com/en-us/library/dn642081(v=vs.105).aspx