Can my iPhone 4 back-up transfer to a new iPhone?

So i lost my iPhone 4 yesterday. It is connected with Find My iPhone but it is offline (most likely dead) and so i can't see it's location. I backed up my iPhone on my Mac about two weeks ago so I'm not that worried about losing all of its data. My question is if i were to get a new iPhone 4 would i be able to get the back-up that's on my computer to transfer over to my new one? Would all my stuff on my old phone then be on my new one? And does the same go for newer versions of the iPhone? Can iPhone 4 back-ups go on new iPhone 4S or iPhone 5's? Answers would be greatly appreciated, thanks.

Yes, you can transfer the file from iphone to computer first and then sync them to new iphone
as to how to get files out of iphone, you can use istonsoft iphone to computer transfer (google it if you want) and here is the video guide to show you how to use it