How to temporarily use a 16gb iPhone when all my data was stored on a 64gb iPhone?

The only way for you to transfer data to the older, small capacity phone, is to sync it with iTunes or iCloud and move data manually. In other words, you could sync your contacts and calendar, and if you use iTunes, you could sync photos from the computer to the phone, but you will not be able to restore your backup to the smaller phone, which you have already found out. So things like the Camera Roll, or Messages, could not be moved from one device to the other. From iTunes, you could select certain apps, if there are a few you need on the 16GB phone, but remember you are going to be severely restricted as far as space is concerned, especially after being used to a 64GB device.

Not really. The closest you could come, assuming there is already a sync relationship set up between the 64GB phone and iTunes, and the phone is functional enough to be able to plug it in and back it up, would be to plug it in, use iTunes to 'unsync' your media and remove enough apps to pare down the total data stored on the device to the point that it would fit in 16GB, then back it up and restore that backup to the smaller phone.