A common question I always had with our Google Apps users was, can you shift the .pst storing file from one machine to another say if they got a new machine?
After doing some research I found one person who had tried it so I thought I would give it a go and here are the results and what you need to do to migrate.
On the original machine locate the Google Apps Sync folder and files. C:\Users\%username%\AppData\Local\Google\Google Apps Sync
Here you should see multiple files, you just need to copy the .pst file.
On the new machine, download and setup google apps sync, configure the user and profile and open Outlook.
Once an email arrives shut down Outlook (Don’t leave it syncing!).
Go to: C:\Users\%username%\AppData\Local\Google\Google Apps Sync
It should look something like this (I have made blanks for the users email and username).
Copy the name of the current .pst file in this location and rename the current .pst in this location by adding .old to the end.
Copy over the old .pst file from the old machine and rename it to the name you copied above (effectively replacing it).
This should trick google apps sync into thinking it is the same file.
Now you just need to remove ALL the other files in this location, (not the tracing folder though). There should now be the tracing folder and the renamed .pst file here.
Reopen Outlook, it will rebuild the files you deleted and should then realize the file exists and just make increment changes rather than downloading the entire email file again.
There are a few reasons why you might need to do this, Saving Outlook only data like flags and categories, or purely because the .pst file is very large (In this case I wanted both the file was 18GB!)