You generally would not encrypt a string using the public key of an X.509 directly. Instead you would generate a strong random(of a specific quality) key; use normal symmetric encryption (such as AES) and then encyrpt the string with that. You then encrypt the random key with the X.509.
Consult a good PKI/Crypto book (e.g. http://www.amazon.com/Applied-Cryptography-Protocols-Algorithms-Source/dp/0471117099) as to why (sections on key leakage, bit-flipping, padding and (re)encrypting twice).
If you really insist on doing this -have a look at https://github.com/dirkx/smime-add-encryption-for-recipient/blob/master/smime-add-encrypt/main.c its pkcs7_encode_rinfo function.
x509cert = ... something to read your x509 byte array in.
unsigned char *stuff = "Some zecret string";
int stufflen = strlen(stuff);
EVP_PKEY *pkey;
EVP_PKEY_CTX *pctx = NULL;
assert(pkey = = X509_get_pubkey( x509cert));
assert(pctx = EVP_PKEY_CTX_new(pkey, NULL));
assert(EVP_PKEY_encrypt_init(pctx)==1);
assert((EVP_PKEY_CTX_ctrl(pctx, -1, EVP_PKEY_OP_ENCRYPT
EVP_PKEY_CTRL_PKCS7_ENCRYPT, 0, ri)==1);
size_t eklen;
assert(EVP_PKEY_encrypt(pctx, NULL, &eklen, stuff, stufflen)==1);
ek = OPENSSL_malloc(eklen);
assert(ek);
unsigned char *ek = NULL;
assert((EVP_PKEY_encrypt(pctx, ek, &eklen, key, keylen)==1);
printf("Encrypted blurp: ");
for(int i = 0; i < eklen; i++) {
printf("0x%02X ", ek[i];
};
printf("\n");
exit(0);