Split Libraries, Inability to Update iTunes

Hello. My sister is going to be getting a new phone soon (iPhone 6 or 6S - she currently has an iPhone 5 on 9.1 iOS), but we have some concerns. She uses a (very old) Windows desktop that runs (very poorly) Windows 7, 64-bit (let me know if you need any additional information on it). This desktop is on its last leg, but she cannot get a new computer for another year and a half, at the earliest; she has to get the new phone this May due to our contract terms.

Here's the problem: a few years ago, we set up split libraries on this desktop so that my sister and my mother could have their separate libraries on the same computer (mom's desktop BSOD'd permanently). The split libraries thing works pretty alright, but - my sister discovered a few months ago that she can no longer update iTunes. This is iTunes 11.4.0.18. She's tried uninstalling/reinstalling iTunes already, but it hasn't worked. We think that the problem lies with the a) old computer and b) split libraries, but she is unable to get a new computer, and mom can't get a new one for a while either.

Sister is able to back up the photos on her iPhone 5 to the desktop with no problem using Dropbox but not syncing the phone in iTunes. If she tries to sync, iTunes pops up a box saying that she needs to update iTunes, which, as I've discussed, seems impossible. We can't un-split the libraries, because then mom will lose her music/apps/videos (I'm assuming?). Sister has been updating her phone through WiFi, which seems to work well, so there aren't any issues with the phone. We've discussed using iCloud to back up her phone, and she is totally fine with that (and she only has about 15gb of stuff on her current phone, so it's not expensive).

So, the big question is: what happens when she gets the new iPhone? Like I said, she can't put off getting the phone until she gets a computer. When she gets the new iPhone, we'll populate it with her iCloud backup (likely while in the Apple store, since that worked well for me). She has a lot of music from CDs, though, and she would love to have those on the new phone as well, but it seems like she won't be able to transfer anything from the computer to her phone, since that link doesn't work now.

Tl;dr: split iTunes libraries, iTunes won't update, because iTunes won't update phone won't sync, have to get new phone before able to get new computer, what do we do?

Thank you so much! I know that this is kind of convoluted, so let me know if you need clarification or any additional information.

ETA: iTunes version.