Not enough memory to back up

If you turn on all the "On/Off" switches on the iCloud settings page, that will transfer all that content over to the iPhone 5 when you sign into iCloud.
In terms of photos, if you don't have enough storage space to backup to iCloud, you will need to email them (This can be done 5 at a time) to your self, and save them onto the iPhone 5 from the emails.

An alternative is paying for more storage on iCloud.

iPhones are not intended to be sole storage of content.

I understand your old computer crashed, but relying solely on an iPhone that can easily be lost, stolen, or damaged beyond repair is not smart.

You do realize that if the device were lost, stolen, or damaged beyond repair that all your content would be lost, correct?

Fix the real issue, get all your content onto your computer. https://discussions.apple.com/docs/DOC-3991

Once that is done, the iPhone can be updated via iTunes without any concern about lost content.

Also, while you can email yourself the pictures from the device, depending on how many pictures and videos are on the device, this could take an eternity. The simple way is to copy the pictures from the device to the computer as you would with any digital camera. This in no way involves iTunes. Importing personal photos and videos from the iPhone to a computer

Couldn't agree more.

I was only replying in the assumption that there was no computer connectivity available 🙂