Switching from Windows to Mac temporarily

I'm a PC (Windows 10) user. My PC's hard drive is failing and I can't repair it until next month. In the meantime, I got a MacBook Air from my company for temporary use.

I want to back up my iPhone 6 Plus and sync it with my iTunes library. But the MacBook Air keeps telling me there's not enough space. My iTunes library is on an external drive and iTunes sees it but won't sync with it or back up my phone. I know for a fact the external drive has more than enough empty space. I can't sync to the cloud because it's only 5 GB (my full iTunes library is something like 300 GB).

To compound the issue my iPhone contract ends on the last day of this month, so tomorrow I want to order an iPhone 7 Plus but if I can't back up my 6 Plus, I won't be able to sync my 7 Plus.

Someone told me a Windows iTunes library won't sync with a Mac version of iTunes, but that sounds odd to me.

So... am I missing something here?