AES S-Box: How is value for 01 mapped to 7c? If irreducible polynomial $m(x) = x^8+x^4+x^3+x+1$ is chosen, or even for any other value, the multiplicative inverse will not exist for $01$, as $0000 0001$ will perfectly divide $m(x) = 100011011$ leaving remainder zero.

So then how it the value for $01$ mapped to $7c$ in AES S-box?
$00000001$ is its own inverse in the Rijndael field, because polynomial multiplication by itself gets $00000001$ and is unchanged by the modulo operation.  This will always be true in any ring, as noted in a comment above.

However, the specification of the Rijndael S-box is that after taking the inverse of the number in the Galois field, you have to multiply the result by a matrix then add (XOR) a constant: the "affine transformation".  That is what makes 0x01 map to 0x7C.

Note that for the purposes of the Rijndael S-box, zero's inverse in the field is considered to be zero.  The affine transformation is applied to zero, getting the expected 0x63.