Try this. First, download this superboot file, extract it into a folder and put it on your home directory.
r3-galaxynexus-superboot.zip
Plug your phone into the computer and put it in fastboot mode by pressing the up and down volume keys and power button at the same time. Open up terminal, then type cd and then spacebar and then drag the folder from your home directory into terminal. It should look like this...
cd /Users/(username)/r3-galaxynexus-superboot
then press enter
Next, type in the follwing code, pressing enter after each line
chmod +x install-superboot-mac.sh
./install-superboot-mac.sh
the terminal should now say:
downloading 'boot.img'... OKAY
booting... OKAY
you should now see a black screen that says google. Your phone should automatically boot up, but if it doesn't after 5 minutes, unplug the phone, take the battery out and turn it back on. You should now have superuser app in your app drawer, and have root access.
After you're sure you have root access, you should install clockworkmod recovery. Follow the directions here on how to do so
[HOW-TO] CWM on LTE Version through Rom Manager - RootzWiki