As the title says, I have some code that generates a pair of RSA keys. I want to split them apart and use them individually to encrypt/decrypt, rather than use the variable
You can extract the RSA public key from RSA keypair using d2i_RSAPublicKey and i2d_RSAPublicKey (link). Use i2d_RSAPublicKey to encode your keypair to PKCS#1 RSAPublicKey stucture, store it in a bytestring, then use d2i_RSAPublicKey to decode it back to RSA key struct.
I want to split [the keypair] apart and use them individually to encrypt/decrypt
... Any ideas on how to separate the keys, then maybe put them back together in a "keypair" or maybe how to use them separately to encrypt/decrypt some string variables?
You can use RSAPublicKey_dup and RSAPrivateKey_dup, without the need to round trip them by ASN.1/DER or PEM encoding them. Here, round tripping consist of using a functions like PEM_write_bio_RSAPublicKey and PEM_read_bio_RSAPublicKey.
Below is a sample program that uses them. Its written in C++ (thanks for adding that tag).
While you can separate them, they both use the RSA structure. The public key has some members set to NULL, like the private exponent.
You can also print the keys with RSA_print, RSA_print_fp and friends. See RSA_print (3) or its use below.
// g++ -Wall -Wextra -std=c++11 -stdlib=libc++ -I/usr/local/ssl/macosx-x64/include \
//     t.cpp -o t.exe /usr/local/ssl/macosx-x64/lib/libcrypto.a
#include <memory>
using std::unique_ptr;
#include <openssl/bn.h>
#include <openssl/rsa.h>
#include <cassert>
#define ASSERT assert
using BN_ptr = std::unique_ptr<BIGNUM, decltype(&::BN_free)>;
using RSA_ptr = std::unique_ptr<RSA, decltype(&::RSA_free)>;
int main(int argc, char* argv[])
{
    int rc;
    RSA_ptr rsa(RSA_new(), ::RSA_free);
    BN_ptr bn(BN_new(), ::BN_free);
    rc = BN_set_word(bn.get(), RSA_F4);
    ASSERT(rc == 1);
    rc = RSA_generate_key_ex(rsa.get(), 2048, bn.get(), NULL);
    ASSERT(rc == 1);
    RSA_ptr rsa_pub(RSAPublicKey_dup(rsa.get()), ::RSA_free);
    RSA_ptr rsa_priv(RSAPrivateKey_dup(rsa.get()), ::RSA_free);
    fprintf(stdout, "\n");
    RSA_print_fp(stdout, rsa_pub.get(), 0);
    fprintf(stdout, "\n");
    RSA_print_fp(stdout, rsa_priv.get(), 0);
    return 0;
}
Here is the output:
$ ./t.exe 
Public-Key: (2048 bit)
Modulus:
    00:aa:5a:cc:30:52:f1:e9:49:3d:a6:25:00:33:29:
    a6:fa:f7:53:e0:3c:73:4c:91:41:66:20:ec:62:1f:
    27:2a:2a:6c:0f:90:f8:d9:7e:d5:ec:72:7b:38:8c:
    ca:12:60:f8:d1:fb:f2:65:7c:b1:3a:b6:4e:26:ba:
    5b:86:cc:30:f2:fc:be:c3:a2:00:b9:ea:81:fa:1c:
    22:4e:f7:be:a1:1a:66:90:13:b6:12:66:26:23:6d:
    22:15:7d:3b:a4:99:44:38:fa:1c:70:63:4e:50:6f:
    66:38:6c:f6:1a:13:e1:c7:dc:a6:a1:eb:6f:f9:c9:
    59:c8:30:dc:c2:1b:dc:6c:9d:ea:0c:3d:52:5a:00:
    ea:c9:c9:85:51:21:9f:ec:95:b3:dc:c2:50:21:29:
    c2:64:6c:1e:34:36:d8:61:59:ab:3c:a2:cc:e8:ef:
    57:c3:7f:49:86:be:e3:42:88:1b:39:10:b8:2f:fa:
    81:ef:a0:94:99:0c:71:ae:1e:82:7f:e3:6e:00:6e:
    02:13:66:bb:a9:31:58:ec:90:39:9c:bc:9c:8c:90:
    e9:20:f7:20:8e:d6:a3:a3:df:a2:4a:0f:0f:39:b5:
    57:b9:ef:6a:27:e0:1a:ed:f6:ce:0d:87:cd:43:03:
    bf:67:ef:ff:fd:da:98:cc:22:ab:5e:8d:7b:43:d3:
    90:4d
Exponent: 65537 (0x10001)
Private-Key: (2048 bit)
modulus:
    00:aa:5a:cc:30:52:f1:e9:49:3d:a6:25:00:33:29:
    a6:fa:f7:53:e0:3c:73:4c:91:41:66:20:ec:62:1f:
    27:2a:2a:6c:0f:90:f8:d9:7e:d5:ec:72:7b:38:8c:
    ca:12:60:f8:d1:fb:f2:65:7c:b1:3a:b6:4e:26:ba:
    5b:86:cc:30:f2:fc:be:c3:a2:00:b9:ea:81:fa:1c:
    22:4e:f7:be:a1:1a:66:90:13:b6:12:66:26:23:6d:
    22:15:7d:3b:a4:99:44:38:fa:1c:70:63:4e:50:6f:
    66:38:6c:f6:1a:13:e1:c7:dc:a6:a1:eb:6f:f9:c9:
    59:c8:30:dc:c2:1b:dc:6c:9d:ea:0c:3d:52:5a:00:
    ea:c9:c9:85:51:21:9f:ec:95:b3:dc:c2:50:21:29:
    c2:64:6c:1e:34:36:d8:61:59:ab:3c:a2:cc:e8:ef:
    57:c3:7f:49:86:be:e3:42:88:1b:39:10:b8:2f:fa:
    81:ef:a0:94:99:0c:71:ae:1e:82:7f:e3:6e:00:6e:
    02:13:66:bb:a9:31:58:ec:90:39:9c:bc:9c:8c:90:
    e9:20:f7:20:8e:d6:a3:a3:df:a2:4a:0f:0f:39:b5:
    57:b9:ef:6a:27:e0:1a:ed:f6:ce:0d:87:cd:43:03:
    bf:67:ef:ff:fd:da:98:cc:22:ab:5e:8d:7b:43:d3:
    90:4d
publicExponent: 65537 (0x10001)
privateExponent:
    66:a4:ce:e3:4f:16:f3:b9:6d:ab:ee:1f:70:b4:68:
    28:4f:5d:fa:7e:71:fa:70:8b:37:3e:1f:30:00:15:
    59:12:b6:89:aa:90:46:7c:65:e9:52:11:6c:c1:68:
    00:2a:ed:c1:98:4d:35:59:2c:70:73:e8:22:ed:a6:
    b8:51:d0:2c:98:9d:58:c3:04:2d:01:5f:cf:93:a4:
    18:70:ae:2b:e3:fc:68:53:78:21:1d:eb:5c:ed:24:
    dc:4d:d8:e2:14:77:46:dd:6c:c5:4b:10:a4:e6:7a:
    71:05:36:44:00:36:ca:75:e8:f1:27:2b:11:16:81:
    42:5e:2e:a5:c6:a3:c9:cd:60:59:ce:72:71:76:c8:
    ca:ba:f0:45:c3:86:07:7b:22:20:c4:74:c6:a8:ab:
    7c:2c:f8:de:ea:25:95:81:79:33:54:67:7b:61:91:
    80:a8:1f:4c:38:32:d4:4d:2e:a8:7d:9b:d4:1a:3e:
    6b:ca:50:3c:a0:61:0e:00:ad:f4:5c:0f:26:1a:59:
    00:3c:bd:ee:c3:e8:d0:b8:9b:0e:44:89:49:d1:24:
    a4:39:15:dc:0e:c5:d5:41:a2:4a:f4:e5:e3:23:c7:
    98:8a:87:f7:18:a6:e2:7b:27:83:f6:fb:62:42:46:
    ae:de:ba:48:ad:07:39:40:da:65:17:d1:d2:ed:df:
    01
prime1:
    00:dd:dc:70:b5:70:ea:10:20:28:40:a0:c3:b8:70:
    6d:3d:84:c0:57:2d:69:fc:e9:d4:55:ed:4f:ac:3d:
    c2:e9:19:49:f0:ab:c6:bd:99:9e:0f:e5:a4:61:d4:
    b3:c5:c2:b1:e4:3a:10:ff:e6:cd:ce:6e:2d:93:bc:
    87:12:92:87:7c:d3:dd:bc:32:54:9e:fa:67:b1:9d:
    e2:27:53:e6:03:a7:22:17:45:63:0d:42:f3:96:5d:
    a3:e0:9c:93:f0:42:8b:bb:95:34:e6:f6:0b:f7:b6:
    c5:59:a0:b5:2a:71:59:c0:f2:7e:bf:95:2d:dd:6d:
    94:23:2a:95:4a:4f:f1:d0:93
prime2:
    00:c4:91:6a:33:1b:db:24:eb:fd:d3:69:e9:3c:e2:
    a2:2d:23:7a:92:65:a8:a0:50:1d:0a:2b:b4:f0:64:
    e4:40:57:f3:dc:f7:65:18:7d:51:75:73:b9:d6:67:
    9b:0e:94:5f:37:02:6c:7f:eb:b9:13:4b:bf:8e:65:
    22:0b:2c:c6:8d:2a:a2:88:ec:21:e3:f9:0b:78:b4:
    1d:d0:44:e6:36:0d:ec:3c:8f:0a:c7:3b:0d:91:65:
    b7:de:a3:c9:a3:2a:8c:7f:1f:a1:d2:6e:9b:ee:23:
    78:c1:30:76:87:af:a8:11:a4:15:b4:54:16:d8:94:
    71:5c:64:30:43:58:b5:07:9f
exponent1:
    2f:91:e8:88:be:e1:30:fb:f4:25:87:52:ef:e5:0b:
    47:39:83:94:2d:a4:a0:19:f2:f1:49:a4:df:a5:8e:
    79:34:76:ea:27:aa:c1:54:82:d3:9d:c5:95:44:6a:
    17:69:1b:83:77:ff:d5:1e:c3:da:13:3d:aa:83:ad:
    e2:89:90:8b:6f:52:07:dc:32:d0:b3:98:30:39:4e:
    18:68:a0:d4:ff:ad:0b:98:51:18:b2:d6:4f:d3:5c:
    23:f8:ee:af:81:55:3c:af:4d:5c:88:3d:20:ac:0b:
    bc:9f:fc:b8:50:fd:91:a5:6d:0f:df:08:aa:85:a8:
    51:b1:fb:b8:a7:53:8e:09
exponent2:
    7d:46:0b:7f:ad:06:19:de:c8:b2:7e:f2:25:5a:6e:
    6f:04:08:6e:da:99:00:2a:6e:87:77:d9:65:c7:76:
    ec:46:e1:64:f6:ca:18:34:6d:c0:c3:d3:31:00:70:
    82:77:2e:c3:59:29:1a:d1:78:ef:02:3c:7f:9c:96:
    78:b6:bd:87:64:1f:97:d1:9d:bb:b3:91:8b:08:87:
    63:9f:35:74:47:a5:41:e7:0b:c0:73:33:2f:71:bb:
    20:0a:14:4c:87:a6:68:b2:19:28:8a:53:98:0e:45:
    3c:22:0d:b8:65:cb:60:0a:c9:c6:56:3d:05:24:7d:
    a6:9b:37:63:04:5a:c3:13
coefficient:
    00:cc:d7:5c:e6:0e:7b:79:d4:cb:4f:6d:82:a7:45:
    90:67:90:dc:d3:83:62:f1:4b:17:43:5c:4a:ea:bf:
    38:25:c3:6f:34:e2:05:91:5e:60:d6:de:6d:07:1a:
    73:71:b3:1d:73:f2:3c:60:ed:ec:42:d4:39:f8:a4:
    ae:d5:aa:40:1e:90:b1:eb:b1:05:a3:2f:03:5f:c6:
    b7:07:4c:df:0f:c4:a9:80:8c:31:f5:e2:01:00:73:
    8a:25:03:84:4e:48:7a:31:8e:e6:b8:04:4c:44:61:
    7d:e4:87:1c:57:4f:45:44:33:bb:f3:ae:1c:d2:e1:
    99:ed:78:29:76:4d:8c:6d:91
Related, you never ask how to encrypt or decrypt with a RSA key, yet you claim the answer to the encryption and decryption problems is shown in the code in your answer.
You seem to have made the question a moving target :) You should probably avoid that on Stack Overflow (I think its OK to do in those user thread forums). On Stack Overflow, you should ask a separate question.
Reviewing your code, it appears that you successfully separated the public and private keys into the strings pub_key and pri_key, but then used printf to output them which pasted them back together. To just print the public key change the printf statement to:
printf("\n%s\n", pub_key); 
I've found a solution to my question among other Stack-Overflow posts and namely Reading Public/Private Key from Memory with OpenSSL
The answer i was looking for is answered by @SquareRootOfTwentyThree is his last line of code,
After extracting the Public key into a BIO variable called pub:
PEM_write_bio_RSAPublicKey(pub, keypair);
i can send the variable pub across the network, and after it reaches the other side create a RSA variable and put pub inside it:
SOLUTION:
RSA *keypair2 = NULL; 
PEM_read_bio_RSAPublicKey( pub, &keypair2, NULL, NULL);
After i've done this i can successfully encrypt the message as usual, using keypair2:
ENCRYPTION:
encrypt = (char*)malloc(RSA_size(keypair));
int encrypt_len;
err = (char*)malloc(130);
if((encrypt_len = RSA_public_encrypt(strlen(msg)+1, (unsigned char*)msg,
        (unsigned char*)encrypt, keypair2 ,RSA_PKCS1_OAEP_PADDING)) == -1) {
    ERR_load_crypto_strings();
    ERR_error_string(ERR_get_error(), err);
    fprintf(stderr, "Error encrypting message: %s\n", err);
}
I can then send this encrypt variable back to the first machine, and decrypt it as usual, using my original keypair, without having to send it over the network.
DECRYPTION:
decrypt = (char*)malloc(encrypt_len);
if(RSA_private_decrypt(encrypt_len, (unsigned char*)encrypt, (unsigned char*)decrypt,
        keypair, RSA_PKCS1_OAEP_PADDING) == -1) {
    ERR_load_crypto_strings();
    ERR_error_string(ERR_get_error(), err);
    fprintf(stderr, "Error decrypting message: %s\n", err);
}
Thank you everyone for contributing to this post!!!