I have 4.0.4 on my Galaxy Nexus with Apex ROM, but I believe it's because I flashed AOKP maguro milestone 5 which takes you up to 4.0.4, and when I flashed back to Apex, it retains the ICS version.
So, I guess you could try that if you are totally stuck on Apex ?? Nandroid your current setup, then flash AOKP. Nandroid that too if you want, then flash back to you nandroid-ed Apex rom. Or stick with AOKP. Both are fantastic ROM's but I get better battery life with Apex.
I am guessing as to why this is so, but I have flashed so many different ROM's on different devices in the last couple weeks, it's all blurry.
I got both AOKP and Apex from another site. Check out GalaxyNexusRoot site
(not sure about linking other sites)
Both those ROM's should be easily found. Apex seems better on battery, but I believe AOKP has the "fast charge USB" option which allows for roughly 2 hour charges from a dead battery to full/near full, and AOKP seems like it's a little more customizable.
Make sure to pick proper type: Sprint/Verizon/GSM
Gummy has good reports as well. There's so many out there, it's simply a matter of trying, and picking.