No, I do not think that most Americans know what the Eleventh Amendment is without looking it up, because I'm an attorney and even I had to look it up. Most people do not even know who the vice president is. The constitution is something that most people are taught to ignore these days, not to study.
Most Americans do not know the contents of the constitution and its amendments. Only a few, such as the right to bear arms, are well-known. Past that, most of the amendments fly under the radar. As long as it does not directly affect their lives, most people are content to go on not knowing.
Most Americans don't even know what a majority of the first ten amendments to the constitution entail, much less the next 17. Even I had to look up what the Eleventh Amendment enumerates even though it was the first amendment ratified by the states after the original Bill of Rights.
I do not believe most Americans know what the Eleventh Amendment is without looking it up. I just looked it up myself and it is not particularly something I would bother to remember. It applies to so few people in the United States that most people simply are not affected by it.