Is that possible to calculate modular inverse of a point on elliptic curves?
No, this is a known variation of the computational Diffie-Hellman problem, called Inverse computational Diffie-Hellman (InvCDH), according to which on input $g$, $g^{x}$, it is difficult to find $g^{x^{-1}}$.

See section 2.2 at Variations of Diffie-Hellman Problem by Bao et al. for more info and problem reduction proofs.