On second thought, side by side wouldn't be a fair test since my tt pre amp is a cheap one.
It may be cheap, but using only one for the test eliminates a variable, so the test will be as fair as possible without buying a more expensive one.
The fact that you posted about it being cheap begs the question- what improvement do you expect to hear from one cartridge vs the other? Also, if you're going to remove one cartridge and compare it with another, you'll need to set up the second one before playing it because a mis-aligned cartridge is a bad way to do a listening test. This takes time and it eliminates your memory of the first cartridge's sound to be totally accurate. Listening to one recording isn't going to show everything that any cartridge can do, unless it's a pristine copy and/or a great test record.
If you do this, I would avoid worrying about "Is this the best I can have?" and listen with the knowledge that they WILL sound different. Once that's out of the way, you have the choices of listening to enjoy or listening to analyze. If you can enjoy the sound quickly, you win- constantly thinking about what you're listening through/with takes away from enjoying the music, IMO.
I don't know about you, but I don't like to think about the equipment while I listen- when I do that, it's to hear the details in order to make a decision about whether I want to sell it but what I consciously do is not care whether it sounds better or worse. Caring may be sharing in other aspects of life, but in a comparison like this, I find that it intrudes on my ability to listen only to the sound. Knowing what is being used intrudes on the listening.