iPhone backup question?

So I've been having a tough time being able to get my iPhone 4S to be detected on my own computer, and I'm at the point where I'm just going to backup my iPhone 4S to a different computer. The thing is, I've recently received the new iPhone 5S, but I want to make sure I have all my data and information backed up before I use my new phone. So is it possible to back up my iPhone 4S onto a different computer than I would usually use, restore that backup onto my new iPhone 5S, and then backup my new iPhone 5S (with the backup/data from the 4S backup) to my own computer?

Thanks so much for your time.

Economicrex662014-04-30T15:31:59Z

All your downloads are already synced to your iPhone 5S. If you mean pictures and media, if you can't put the media in the computer, put it in another computer. I not then you can't save any of the pictures from that phone. Sorry.