Is there a partially homomorphic quantum secure public key cryptosystem with IND-CCA1 security?
Essentially any IND-CPA-secure lattice-based cryptosystem offers additive homomorphism, up to a predetermined number of operations.

I don't know of any IND-CCA1-secure post-quantum candidate that offers any homomorphic property, except Loftus-May-Smart-Vercauteren SAC'11, which is based on a nonstandard "knowledge of error" lattice assumption.