I would like to encrypt a string asymmetrically.
I would like the ciphertext to be as short as as plaintext. (or shorter)
I've read about format preserving encryption functions - are there asymmetric variants of them?
Yes, in fact there is at least one length-preserving asymmetric encryption scheme. It's deterministic, though, so there are some security tradeoffs that come with it. It's described in the paper "Deterministic and Efficiently Searchable Encryption" by Boldyreva et al. Look for "RSA-DOAEP".