iPhone 4s became hot quickly draining battery and shutdown unexpectedly
It sounds like a battery issue.

You may need to take it to a repair shop and have it replaced IMMEDIATELY. 

I say that because I have seen an iPhone 4 battery explode before. It's not a pretty sight. Consider the average lifespan of a lithium battery: 


  The typical estimated life of a Lithium-Ion battery is about two to
  three years or 300 to 500 charge cycles, whichever occurs first.


This is a very common issue for the iPhone 4 and can be repaired pretty easily. 

I'm also going to cite my personal and proffesional experience with this.