Thanks. I tried it and get the same output, read only filesystem..
EDIT:
Ok so I did a "remount" and then managed to push the zip to /system/sdcard.
But when I went to recovery mode and choose "apply update from /sdcard" the zip is not there (or any other files..). What am I missing?
It's been a while since I've messed with my Nexus' file system, but /system/sdcard is not a standard directory. When the device is operating in its normal mode, the SD card is at /mnt/sdcard, which mounts the /sdcard directory.
You could try checking things out with 'adb shell' then you can do an 'ls' and take a look at what your file structure looks like, using 'cd' to go into different folders. adb push should have root priority, so it's odd that it wouldn't let you copy a file to /sdcard.
Another thing you could try would be flashing the Clockwork Mod recovery image. You'd just power your phone onto the bootloader (the Volume Up + Power button), download the latest Clockwork Mod for your phone, put it in the same directory (on your computer) as the fastboot.exe, and then do "fastboot flash recovery [your Clockwork Mod file name]"
The Clockwork Mod recovery has USB mounting features that would let you copy files to and from your internal storage. If you can successfully get the Clockwork Mod recovery on your phone, you could actually flash any custom ROM (intended for you phone) that you wanted, but to try to fix things, it's generally best to flash to an official Full ROM version for your phone.
Considering that your phone appears to be having issues copying things, I'd recommend saving the Clockwork mod thing for last. As long as you can get into the bootloader as normal, it shouldn't be a problem though.