Can I backup my iphone 4 to a new computer?

I have an iphone 4 and I'm going to buy and iphone 5s. However, I have a new laptop and everything is backed up on my computer at home which is currently broken.SO, can i get an iphone 5s and still recover all my photos and contacts? I'm worried too much about music and apps. Just contacts, text messages/history, and pictures/videos.

Just download Itunes, and syncronize trough Icloud.
OR just make a separate safety copy of the old one.

If you can back it up on iCloud than that would be the easist way to go.however, you can also backup your iphone 4 to this new laptop as long as you didn't backup your phone on more then 5 different computers. If you have then you would have to go to itunes and delete the rest of the backups and then only you can back it up again. Also if all you are concern about is photos and contact then any number of backups can be done but only upto 5 computers can hold the data of your apps, movies and music.

If you have iCloud, it would be much easy to transfer files from iPhone 4 to iPhone 5s, no need to backup.