For RSA cryptography, how long does it take to factor out P -1024 if given Q - 1024 and N - 2048? So doing a little research on RSA, and knowing the public key contains N. So breaking down the steps to cracking it, I'd have to search for P and Q (which takes a very long time as I understand it), (where P*Q = N). aka, prime 1 and prime 2.

So it takes a long time to find a suitable prime pair. For me I see 2 actions and I want to know which action takes the longer time.


Finding a 1024 prime number.
Having 1 prime number and dividing N by testable prime number to see if it produces another prime, aka prime 2.


So is the time mostly spent finding prime 1 and THEN trying to factor out N, or is the time spent finding a large prime, but its easy to divide N by prime 1 and see it not work?

Sorry if this is confusing, I am having a hard time constructing the question.
You seem to be under the impression that factorizing (finding P and Q) means to try many primes and hope to find the right one, aka the "brute force" solution. This is not the case, it would take forever. Eg. 1024bit means  


  179769313486231590772930519078902473361797697894230657273430081157732675805500963132708477322407536021120113879871393357658789768814416622492847430639474124377767893424865485276302219601246094119453082952085005768838150682342462881473913110540827237163350510684586298239947245938479716304835356329624224137216


possible numbers. Even with the best computers, it would take far longer than the universe exists. And to start with, 1024 is in the past, nowadays 4096 is common.
...point is, there are better (and more complicated) factoring methods, and with quantum computers it gets much better again (sad for cryptographers :/).  

Anyways, to answer your direct question: "Neither".
"Finding" a prime number actually means  


  Take any number and run some checks if it could be prime. If the check says
  "could be" often enough, then let's say it is a prime.


Why? Because getting hard evidence that a 4096bit number is prime is very slow too.  

So, finding a prime means running checks. Checking if the second number (after division) is prime means running the same checks. There is no difference other than the division.