Given a ciphertext $c$ and a public string, how do we check whether $c$ encrypts something that is roughly equal to the string without decryption? Let $$ be an encryption function, then we have $c = f(m)$ for some bit string $$. Here, we also have pattern matching function $(,′)$ which returns true whether $$ matches roughly with $′$ in terms of some criteria. Given $$ and $′$ (some public string), we want to clarify whether $$ encrypts $$ such that $(,′)=$ without decrypting it.

How can we do that?
I'm not looking for solutions based only on homomorphic encryption since the result will never be revealed without decryption. 

Does ZKP help this?
In short: ZKP does not help mostly if you keep your scheme like that. But your scheme can be fixed if you don't need to rely on public encryption.

ZKP do help if a problem can be efficiency calculated if a witness is known and otherwise it is hard to compute. And that's the problem here since anyone can generate such proofs in some cases.

If we talk about encryption schemes, there will be three kind of schemes:

(1) One-Time based: An adversary chooses any m which g(m, m') returns true and compute $k = m \oplus c$. Now anyone can create the desired proof.

(2) Symmetric-key based: An adversary randomly chooses a key k and computes $m = f_k^{-1}(c)$ and outputs k, m, if g(m, m') returns true and otherwise repeats with a different k (algorithms used in practice might have simpler attacks too). Especially if the pattern function has a wide range the likelyhood to find a true returning pattern will break the ZKP since anyone can could generate such a prove then.

(3) Public-crypto: I'm not sure myself here, but I would not rule out attacks like in (2).

The attacks I described here works because the key can be chosen by an adversary. If you rely on the random oracle model, you can use a hash function as commitment (e.g. SHA256) to commit your key using $com \leftarrow H(k|decom)$. Now an adversary must generate a prove so that he knows a key k, a message m and a decommitment decom so that $H(k|decom) = com \land f_k(m) = c \land g(m, m') = 1$ which is hard to compute if you don't know the key and the message.

I would recommend you to use a generic  zero-knowledge protocols which proves the computation of a boolean circuit like presented here and here. The authors first paper even published a reference implementation of SHA256 on github. Hopefully your encryption scheme does not rely on public encryption, since otherwise the proof becomes impractical.