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?
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/showSoftwareDownloads.do, and under "Wireless Data Software Applications," you'll see iDEN USB Installation Application/USB Modem Driver. Click on the link and you'll be taken to a page where you can download this application, depending on whether you have 32-bit or 64-bit Windows. Click "download", then "Open." It'll save to your hard drive as a .zip file. Then right-click "Uncompress" or something along those lines and it will fully install to your hard drive. There will be a .exe executable file, just "Run" that and it will install the USB drivers you need for Windows to recognize your i335.
Next, relating to your particular concern about backing up contacts info, on that main page http://servicemyphone.motorola.com/iden/support/showSoftwareDownloads.do up above, there's an application called iDEN Phonebook Manager. Pretty much the same thing as far as installation goes, there are installation instructions for these apps.
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.