I have a salted hash (say about, 110 characters long) and the corresponding plain text that I used for hashing. As the title says, is it possible to do find the algorithm via crypt-analysis? What is the recommended approach, if it exists?
Yes and no.
Yes. If by cryptographic hashing algorithm you mean (an unkeyed version of) one of the well known ones, then you can simply try all of them. There's not too many of them.
No. If by cryptographic hashing algorithm you mean just any such algo, then no. There are infinitely many of these.
(And you can trivially generate new hashing algorithms, simply by turning one of the unkeyed hashes into a keyed variant by plugging that hash and a random key into the HMAC construction for example. They will then behave like a completely new hashing algorithm.)