The Brooklyn theme will automatically place your logo in the center of the screen if your main menu is long enough to be collapsed down to a menu on the left side:
Outside this native automatic positionning, you would need to edit the theme’s code to achieve this manually. A developer here on our forums or a Shopify Expert may be able to advise on how to do this.
If you are open to a new theme, our free theme Crave would allow you to position the logo in the center. If you want to recreate the aesthetic that shows the main menu aligned with the logo, a paid theme like Venue or Symmetry will provide you with this design.
I hope this helps. If you need assistance with anything else, please let me know.