The default setting in this tool is to only sync up 1GB of your mail. But you can easily change this with a small tweak you might not be familiar with. In order to do this, you can right click on the icon that appears in your taskbar for Google Apps Sync. Select "Set mailbox size limit".
From here, you should see a range of sizes. Note that if you do not see this, that is because your locally stored PST file for Oulook is already much larger than any range that is available here.
While it's great to have this option available, keep in mind that as the size of your local Outlook mail grows, the slower the performance becomes. That's because it has to continue to stay synced to your mail that is hosted online. So it is best to keep this amount as small as you can for your recent mail and then just use the web interface as your reference place for older correspondence.
And in terms of the Unlimited setting you see in the screenshot above, Oulook limits the local PST file to about 20GB so you won't really be getting everything loaded locally. But that's close to Google Mail's 25 GB limit anyways - and a lot of mail to manage!