How to get all my apps and data off iPhone 4s to new 5s
Yes, you can. As long as your iPhone 4S and iPhone 5S are using the same version of iOS. So first thing to do: Make sure your iPhone 4S has the same iOS version installed as your iPhone 5S. I would recommend upgrading both of them to iOS 8.

When that is done, backup your iPhone 4S to your iTunes.  After you do that, then connect your iPhone 5S and choose to restore it from a backup. And that backup being the iPhone 4S backup you just made.

The other alternative if you do not have a lot of data is to backup your iPhone via iCloud and restore your iPhone 5S similarly.