Translation:I think they were sincere for once.
Why not "I believe that they were sincere for once"? Believe and think are practically interchangeable in this sentence, with believe being one step higher in certainty (depending on context and inflection).
Is "penser" and "croire" similar in French? Or is there an inversion? (with "penser" being more certain than "croire")
"Croire" may be used with or without the subjunctive. Just as with the English "I believe" it can be used to make a statement of factual belief, or it can be used to imply that you may or may not know something to be true.
Without the subjunctive this sentence is simply a statement of "I believe they were sincere..." (without a tone of implied doubt). Adding the subjunctive would make it into something like "I believe the were sincere, but...."