Is there any way to sync iPhone with Google contacts with custom labels?
Depends on how you sync  Google Contacts to iPhone. If you do it by setting up the gmail account on your iPhone using  the Exchange option, then fields in your Google contacts do not transfer to the phone. But the advantage of using the Exchange way is that syncing is instant--edit a contact on the phone and it changes in your Google account as well. 
If you sync the contacts using the gmail option, the custom fields in Google contacts DO show up on your phone. The disadvantage is that syncing only happens when you sync the phone with iTunes on your computer.
That's my experience, anyway. I would like to know how to have both: sync custom field labels and sync instantly on the fly with every edit.