Is the RSA cipher alike to the Shift/Caesar cipher?
You are close but not exactly. In short, RSA is a trapdoor permutation. 

Let $(n,e)$ is a RSA public key, then $y = f(x) = x^e \bmod n$ is a trapdoor permutation. RSA is a permutation since the function $f:\mathbb{Z}_n^*\to \mathbb{Z}_n^*$ is bijective. 

RSA is a trapdoor permutation. Normally you can find the inverse a permutation if it is given explicitly. In RSA, however, given $x$ and the public key $(n,e)$, we can easily compute $f(x)$. If one is given a $y$ and the public key $(n,e)$, it is difficult to compute $f^{-1}(y)$. If one has the private key (the trapdoor),  can compute $f^{-1}(y)$. 

RSA is not a shift-cipher. Although, a shift-cipher is also a permutation, seeing a randomly generated 2048-bit RSA public key that acts like just a shift-cipher will amuse any cryptographer. Actually, this is not possible with small public exponents.

Like any permutation, for textbook RSA, one can write a table and store it, so that you can revert it very easily. The encryption oracle is free. However, there are problems with this;


Today RSA requires at least 2048 or more bits to be secure. Therefore the table for the reverse permutation is infeasible. You cannot store it, however, you may perform short message attacks if textbook RSA is used. Generate all small short messages and compare them.
Textbook RSA is not used in practice and should never be used. For encryption there are padding schemes like PKCS#1 v1.5 padding or Optimal Asymmetric Encryption Padding (OAEP). The former is hard to implement and has attacks occur, again and again, Bleichenbacher and Manger attack. PKCS#1 v1.5 padding has no security proof, however, OAEP has one. If one ever consider encrypting with RSA should consider OAEP. These paddings have random bits that prevent getting the same result for the same input. Therefore with these paddings, the RSA permutation is randomized.


In practice, RSA is not used for encryption. It can be used for;


digital signatures with Probabilistic Signature Scheme (PSS) 
Key encapsulation mechanisms
Hybrid encryption : RSA-KEM with Data Encapsulation Mechanism (DEM). Composition of a KEM and a DEM provides us the standard of IND-CCA2/NM-CCA2—ciphertext indistinguishability and non-malleability under adaptive chosen-ciphertext attack if an authenticated encryption mode is used like AES-GCM, ChaCha20-Poly1305, or 
crypto_secretbox_xsalsa20poly1305