Are there any strategies to determine bond order and hybridization rankings without drawing the Lewis Structure?
Although it is often taught otherwise, hybridisation is a function of geometry. Therefore you must know (or be able to predict) the geometry. On a basic level, the easiest way to do that are Lewis structures.
A different way is using the VESPR theory, a rudimentary model to predict geometries. Be warned though that this model will fail when it comes to more complex molecules. You need a good imagination of the three dimensions to be able to do it without drawing it.
However you want to spin it, you will have to know which atoms are bonded to other atoms. If you can imagine that, that is good, if not you will have to draw it out. The longer you study, the more present some common molecules are and you will be able to tell certain properties by what we like to call chemical intuition. But that takes time.

Bond order is a bit of a tricky topic, because you have to know the bonding situation. This would include all possible resonance structures for some molecules. While knowing or judging this also comes with experience, it is probably best to draw such structures for learning purposes.

I personally do not like exam questions like these, because they usually just test if a student is good at cramming facts, rather than asking about if these things are understood. But things will get better, if you don't lose interest in the mean time.