Is it possible to recover a private key, given a public key and a substring/a part of the corresponding private key?
Yes, but it depends on how much of the private key you have.
RSA can always be brute-forced if you know the public key; it is protected by the sheer length of time that it will take to try out all possible factors.
If you have part of the private key, this means you can rule out a number of candidates - those that don't contain this part of the private key.
How much of an advantage this is, depends on how big the public/private key pair is, and how much of the private key you have.