You can root any baseband version of the Sprint GS3. I rooted on LI3. The link you posted will walk you through it.
To summarize the instructions:
1) Download all the files
2) Place CWM_SuperUser_v3.0.7.zip on your phone (not an external SD card)
3) Extract Odin.
4) Put the clockwork.tar file in the odin directory.
5) Install the drivers (if you haven't already, the link is in that post you linked)
6) Boot the phone into download mode (turn off, then hold vol down, home, then power)
7) Follow the prompt to accept that you know that you can screw up your phone if you do something wrong.
8) Open Odin, connect your phone to the computer
9) Sit tight while drivers install. You'll know when it's done when Odin says COM## (where ## can be any number, even single digits).
10) When you see com##, click on PDA and find clockwork.tar.
11) Uncheck Auto reboot. If your phone reboots into Android, you have to do this all over again.
12) Click start
13) Wait for it to finish. It'll say DONE above the COM##
14) Unplug and pull out the battery after DONE. There is no timelimit.
15) Replace the battery and go into recovery (vol up, home and power)
16) Flash CWM_SuperUser_v3.0.7.zip
17) Reboot. Verify root is set up correctly by open Superuser and swipe to the left and update the binary. You may have to update the Superuser app in the Play store before this is available.
