How can I use the NP complexity Venn diagram to quickly see which class of NP problem can be poly reducible to another class?
As Raphael said the definitions are important, especially if you try to create proofs. These definitions are not always captured in the Venn diagram. 

The definition of NP-complete is not made clear from the diagram and will not help you to reason about 


  "If A is an NP-complete problem, $B$ is reducible to $A$, then $B$ is...". 


You cannot see from the diagram that this must mean that $B$ is NP.

To answer your question if it is not possible how you will be able to drill it into your head is much more complicated. But again the first step needs to be the definitions. 

If you know that NP-completeness implies two properties of $A$ by definition, then you can take the next step. 


  What does $B$ reducible to $A$ means when regarding these two properties. Do the properties of $A$ tell anything about $B$?


The second property says that any problem in NP reduces to $A$ in polynomial time. So if $B$ reduce to $A$ in polynomial time then $B$ is in NP.