ElGamal encryption with private key Does the encryption with a private key and decryption with a public key works in ElGamal likewise what is done when we use RSA for sender authentication?
No, that does not work for ElGamal. If you look at ElGamal encryption and signatures you will see that they are fairly different.

That works for RSA because RSA is commutative, i.e., $f_{d,n}(f_{e,n}(m))=f_{e,n}(f_{d,n}(m))$ for all $m\in Z_n$ holds, where $f_{e,n}(m):=m^e \mod n$ is the enryption function and $f_{d,n}(m):=m^d \mod n$ the decryption function. You can easily convince yourself that this holds. 

That is the reason for why there exists the bad practice oft teaching signatures as "encrypting with the private key" for signing and "decrypting with the public key" for verification, since the above holds for RSA and people tend to generalize that. But thats not the general case, and, as said,  is a bad practice.