I used to study at Cambridge School.
I am used to studying at Cambridge School.
If the verb is continuous tense, it need to be added "verb to be" before used to?
This sentence discusses an action that happened entirely in the past.
That would be past continuous.
To be used to something is a special phrase that means being habituated. For example, "I am used to eating spicy foods, so this dish tastes bland."
actually means, "I have a habit of studying at Cambridge School," which doesn't really make sense. It could make sense if you added some details: "I am used to studying at the Cambridge School library, which offers free Wi-Fi access to the academic journals I need. When I do my studying anywhere else, I find it very inconvenient."
Incidentally, if you are talking about the University of Cambridge, you would usually just say "Cambridge":
There is no continuous construction in either of these sentences. Rather, the used to collocation has two distinct meanings, taking two distinct sorts of complement.
Here used to is employed as a pseudo-modal verb expressing a past habit or state. Like true modals it takes the infinitive form of a lexical verb as its complement.
Note that this expression is never used in any form or construction except the simple past.
Here used to is employed as a transitive adjective meaning, approximately "accustomed to" or "inured to". It takes a Noun Phrase or Gerund Phrase as its complement.
In B2-type constructions, VERB may be BE, either as the main verb or as an auxiliary—but it has to be in the gerund form.
It is theoretically possible to use BE here in a progressive construction, but it's very artificial, in part because English speakers dislike adjacent -ing forms. I find it difficult to think of a situation in which this would arise naturally.