It's not battery dependent, unless your power saver settings kicked in at some point and turned off WiFi.
You can sync Gmail with either wifi or mobile data; you don't need both on at the same time.
If you want to sync with Outlook you can do this via 'HTC Sync', using it on the computer that Outlook is on. The 'HTC Sync' software is on the SD card of your phone. Install it to the computer via the phones USB cable. It syncs other things besides contacts.
Alternatively, and easier, the Android Sync Manager WiFi app allows you to wirelessly sync with Outlook, as well as other types of files and media. It does more than 'HTC Sync'.
