But what about situations when we mean at some university by saying just at somewhere, for instance, and I've seen a lot of examples in Internet, I want to study at somewhere like *a name of a university* ?
You can only <verb> something at a specific place or places. Anywhere and everywhere are not specific therefore you cannot use at. Somewhere is specific. so you must uses "at". Somewhere = a specific place but I do not know its name.
The words anywhere and somewhere include the concept of "at a place", so we don't use a second "at" with them. Words like "university" don't include the concept, so we need to say "study at a university".