Why is NaNO3 usually used as a salt bridge rather than CuS or NaOH?

NaNO3 or KNO3 do not form any insoluble salts with the electrolytes used.  Most of the salts formed by Na+, K+ and NO3- are soluble. So they do not interfere with the movement of the ions.

No two salts are exactly the same, and there may be specific cases where NaNO3 is the best choice, but most likely this is just your teacher's preference.  NaNO3 is a strong electrolyte?Na+ forms the strong base NaOH which completely dissociates in water, and NO3- forms the strong acid HNO3 which also completely dissociates in water.  But you could also use KNO3 or NaCl, for the same reason.  You mention NaOH, which would be a good choice, but CuS is a less attractive salt because it is not as soluble as NaNO3, NaOH, KOH, NaNO3 or NaCl.  NaOH might not be as good of a choice becaue OH- ion is more likely to form an insoluble compound than NO3-.

So the answer is yes, other salts can and often are used as salt bridges.  NaNO3 is by no means the only effective salt bridge, but it is a very good one.  Sodium and potassium make good cations for salt bridges because they nearly always form soluble salts.  For anions, nitrate and sulfate are good choices.  There are others.

In real life the choice may depend on some unique property of a specific salt and also on the cost and availability of different salts.  For in class examples, you should ask you teacher.  It might make a difference, but it may also be your teacher's preference.  For example I had two different professors for organic chemistry and one always seemed to use KOH as a strong base in examples, while the other appeared to prefer NaOH.  In most if not all of the examples, either one would have worked.