Would it be right to say "I was at that school before," or do I have to say "I went to that school before/I studied at that school before"?
I was at that school before is grammatical. Whether or not you use it depends entirely on what goes before it and what follows it. Seen in isolation it looks like an improbable sentence.
If you just want to reference time, it should be "I went to that school" and just stop there. The past tense already tells you that it happened in the past.