Data stored within your phone's internal storage media is going to stay there no matter if there's a battery or even no battery.
But that's not even something you should worry about, the fact is you really need to set up an automatic backup system for your phone data. Since you apparently don't have your email/contacts/calendar syncing with your email service, it's also likely there are things like photos, music, and other files too that may or may not exist only on your phone. The thing is, smartphones are not intended to be used for long term archiving. If your files and important personal data only exist on your phone, that's a big risk.
So for your own sake, start backing up your phone's user account. Backups are important. They really are. Especially when you send your phone off somewhere to be serviced by a stranger. And be sure you've set up a lock screen.