I think the ANA lounges have some curtained off areas with reclining chairs, don't they? If you want to avoid going through immigration and the day rooms are not available, I think that may be the best spot for a few hours of shuteye.

I can't really conceive of spending 11 hours airside at NRT, but I know that going landside is not an option for some people. That said, I think it's worth trying your luck at immigration. Even if your passport doesn't normally entitle you to be a short term visitor, the immigration officials can allow holders of most passports into the country on a transit visa or shore pass.