Can i download the new iOS without computer internet connection? My super old desktop is currently having problems connecting to the Internet. 

Is there a way that I can update my iPhone to the new operating system without my computer connecting to the internet? 

I'm afraid that if I connect my iPhone to my newish laptop to do the upload that it will delete all of the files and songs from my phone since they are not on the laptop's harddrive. I have not updated my phone in almost a year due to this problem.
If your iOS device is on version 5.0 or higher, then yes, you can do it directly from the handset by going to Settings/General/Software Update.

If you're on iOS 4.x then no, you will need a computer to do this.

To upgrade in your situation, I would recommend the following steps:
1) On your new laptop, open iTunes, and in the preferences on the devices tab, make sure Prevent iPods, iPhones, and iPads from syncing automatically
2) Plug your iPhone in to the new laptop, and in iTunes, select the iPhone on the left pane.
3) Click Check for update.

It should find a newer version, and start to download the file. You can now disconnect your iPhone from your new laptop.

Once it has downloaded this file, you should be able to find it on an OSX system in your home folder/library/iTunes/iPhone Software Updates. On Windows it should be in your user profile folder\Application Data\Apple Computer\iTunes\iPhone Software Updates.

You can copy this file to your old computer, and then plug your iPhone in to that.

On a Mac, hold the Option key (alt on modern keyboards) (shift on Windows) and click Check for update. You should now be able to manually navigate to the ipsw file you just copied over.

It should now start the update process on your iPhone.