I'll assume you're using Windows XP/Vista/7, since if you have a Mac, I'm pretty certain you're out of luck.
What you want is an original Motorola mini-USB data cable SKN 6371, which you can get from Sprint Nextel, ebay, or any of the thousands of cell phone accessory dealers online these days. I straongly suggest original Motorola only.
You'll also want to navigate to http://servicemyphone.motorola.com/iden/support/sh
Next, relating to your particular concern about backing up contacts info, on that main page http://servicemyphone.motorola.com/iden/support/sh
In my experience, you'll want to install the USB Modem Driver first, restart Windows, install any apps (Phonebook Manager, iDEN Media Downloader, whatever), restart Windows, then start the app, and THEN hook up your i335 last. Using Phonebook Manager to copy to your PC is pretty easy. And best of all, your PC can now charge your i335!
Let us know how it goes.
How do you connect this device to your computer, or is it possible to connect it to your computer to back up contact info?
It's possible to connect any iDEN device to your computer. What you will need in 99.9% of the cases is the appropriate data cable specific for your phone, and possibly the appropriate software to read the data from your phone so that you don't have to do a ton of editing. Could a skilled iDEN tech provide the appropriate cable and software info needed to do what the OP is asking please?