Unfortunately, you are very limited in what you can do with that backup. In fact, there is literally nothing you can do with an iCloud backup unless you restore the backup. This can be done, however, on any iOS device (as in iPhone, iPod, or iPad), that is at least the iOS version that the stolen device was on. For example, if the iPhone 4 was running iOS 6, you will only be able to restore to a iOS 6, or 6.1 device.
If you are on a Mac, you should have easy access to Calendars, Contacts, Notes, Reminders, and such data. If you are on a PC, you can access most of that by going to iCloud.com. This isn't really the data that you've backed up, though, and it's really data that is just "synced" through iCloud. Either way, Contacts, are typically pretty important.
If you own a Mac, you should be able to get your Photostream pictures by installing and opening Photostream. This doesn't capture videos, though.
If you do have an eligible iOS device that you can do a restore on, be sure to back up that device locally before wiping. It might also be advisable to make sure that device doesn't do a backup of the newly erased setup to iCloud. If you keep that iCloud account disabled on that device, you should have an easy restore from iCloud, and a backup in iTunes should something go wrong.
Turns out I solved the issue by just wiping my phone and restoring it from that same Sept 27 iCloud backup again. It had the photos in the backup after all. As to why it didn't download them to my phone when I restored from that backup a month ago, I'll probably never know.
Before I actually wiped/restored, I wanted to make sure that the backup actually had the photos in it. Of course Apple doesn't let you see the data in your backups, so I had to download a 3rd party backup viewer on my mac to look at the pictures in the backup file. Once I saw the photos were really there in iCloud, I decided to just restore from that backup and see what happens. It ended up working. Weird.
Best Answer
Google Photos lets you store unlimited photos in high quality on their servers, just by downloading the Google Photos app. The app works great, and I've found it to be reliable. It gets 5 stars in the iOS App Store (which is really quite an achievement) and has nice extra features in the
Assistantsection, like auto-making videos from your photos/videos. Once your photos and videos are backed up to Google Photos, you can view them on phones, tablets, and computers, just by signing in to your account.You can also spend $1 a month to purchase 50 gigabytes of iCloud storage, and turning on iCloud Photo Library lets you back up all of your photos similarly (though I've found this to be slightly less reliable than Google's solution).
Read more about backing up your photos with this New York Times article.