Cfrtjryk

1. 
   

   Be strong on

   
   
   

   
   

   I'm not too strong on organic chemistry, so I'm going to get a tutor this semester.

   
   

   
   


2. 
   

   Be low on

   
   
   

   
   

   We are running low on rice.

   
   

   
   


3. 
   

   Be long on

   
   
   

   
   

   He was short on patience, but long on a sense of his own worth.

   
   

   
   


4. 
   

   Be short on

   
   
   

   
   

   The president’s speech was long on colorful phrases but short on solutions.

   
   

   
   

I think that 'on' is used to convey the meaning of "regarding"

But why is it incorrect to say this way?

5. Be good on
   
   

   
   

   I'm not very good on predictions

   
   

   
   

To my (educated American) ear, sentences 2, 3, and 4 sound natural. All of them have the form "short on <measurable quantity>" or "long on <measurable quantity>".

In sentence 1, "I'm not too strong in organic chemistry" would sound more natural to me. This sentence is about a field of knowledge that you are either good at, or good in.

In sentence 5, "I'm not very good with predictions" would sound more natural to me. Unfortunately, both "on" and "with" are unclear in this sentence: Are you not very good at making accurate predictions, or are you not very good at making use of other people's predictions? "I'm not very good at making predictions" would be both clear and natural.

I agree with Jasper about #1, so for me the question that underlies your question is what is it about strong and good (when they mean "skillful, capable") that makes them unable to partner with on as low and long can?

We are low on fuel.

He is not very long on patience. He is short on patience.

There, on introduces something for which we have a supply. The supply can be ample or it can have become depleted. We can "run out of" patience just as we can run out of fuel.

Skill and capability cannot become depleted. We do not have a supply of skills that runs out when we use those skills.