I believe there are backup tools, but I know nothing about them - a quick google is probably your best bet.
Alternatively, a good place to start would be Settings .. Backup and restore and tick the Backup and Restore options - that should save most of your phone settings.
For calendar and contacts, the easiest way is to sync with Google. You may already be doing this - log into your Google account using the browser and check if everything's there. If anything's missing, go to Settings .. Google (under accounts), tap your account name then tick all the sync options. Also make sure you have sync turned on (easiest way is via the Power widget).
Alternatively, you could sync with something like Outlook on a PC - there are a variety of Sync tools available on the Play Store. I use one called AndroidSync that works over WiFi.
If you've upgraded to 4.4 you should have had an option to back up all of your photos and videos to G+ (I'd do the initial backup over WiFi if I were you).
If you don't have that option - and for all your other files (music, docs etc) - you could either use some cloud service (I think G+ gives you 5GB or so, MS also has something with a few GB for free etc etc) or you could connect to a PC via USB and copy everything there.
Let us know how the repair goes - my N4 was on a wall charger all of last night and the battery was in the red this morning so I'm a bit worried I may be about to have similar issues. Did eventually charge on USB, though.
Oh - and you should definitely do a factory reset before sending the phone away. Google probably already has all your info, but the fellas in the repair shop don't.
