Assuming you are talking about the "rating" of something (in other words a "grade" or judgment about how it compares to other things in its category), then you would use "rating" instead of "rate" in your second example:
'something was given a lower rating'
Your first example sounds fine:
'something is rated lower'
"Lower" is, as you say, an adjective, but that is what you want in this situation. "To be rated" is a so-called "linking verb" like "to be", "to seem", or "to become". A linking verb links the subject ("something" in this example) to an adjective ("lower") which describes that subject. You can read about this type of sentence in this "softschools "article about predicate adjectives.