can you backup on icloud if your phone is disabled

Well I can't guarantee it but there's a reasonably good chance that it will. There are some things that could go wrong. Some users find that there is a time lag between purchasing additional storage and it becoming available. You may also find that it won't back up even with additional storage if there is a problem with your existing backup. You may also not know how much additional storage you need to purchase since you can't get into your phone to see how much space it estimates the next backup will take.

In order to purchase the additional storage you would have to update your Mac to Mountain Lion (available through the Mac App Store), then go into System Preferences>iCloud, sign into your iCloud account, click Manage... on the bottom right, then Change Storage Plan on the top right. This is all necessary just to purchase more storage in your account (since you can't do it on your phone). It's up to you if you want to try it knowing that it may not work.

Of course, If you have an iTunes backup on your boyfriend's computer, you won't need to.

Do you know if you ever set up iCloud on your phone, and if so, if you ever turn on iCloud backup?

What version of OS X are you running on your computer? If you don't know, click on the apple logo on the top left, then click on About This Mac. The version will be listed under the OS X heading.