Method 1 : Eliminating Options
The fastest way would be "Eliminating Options".
Here, RE $II$ can be eliminated by Taking String $10101$..which is accepted by the FA in Diagram But not generated by RE $II$.
So, You can eliminate Options $A,C,D$ as they have RE $II$.
This method can only be used to Eliminate options, Not for Selecting correct Option. This method is based on the quote/expression "When you have eliminated the impossible, Whatever remains however improvable must be the Truth"..
We were lucky that Three Options got eliminated here in this Question. But this would not be the case everytime, So, Knowing correct logic is required.
Method 2 : Sense the Language of the FA :(Best method if we can)
With lots of practice, we can make ourselves able to sense the language of the FA in most questions.
Here, It is a Standard FA which accepts the language "Ending with $1$" . So, Answer would Option B as RE $I$ and $III$ generate the language "Ending with $1$"
Method 3 : Convert the FA into RE using Boring and lengthy algorithms
Knowing every algorithm in the syllabus surely helps sometimes. Here also, You could just use the "FA to RE" algorithm and get the answer. The one drawback of this method is that algorithm will give 1 or 2 RE But maybe none of them are in the options. Here,in our question, It is not the case. So, Apply this method and see.