There is no one “right” way to do things! These resources are an outline of my own approach, but others have many different successful strategies. Ask around and find what works best for you.
A lab manager or research coordinator position is different from an undergraduate research assistantship. Labs may be looking for someone who can contribute to research, but they especially want someone who can keep projects moving forward, work reliably, and solve problems independently. Demonstrating your skills and abilities will get you on the short-list, and being a good research fit will help put you over the top.
Show that you understand what the job actually involves.
A lab manager is not simply a full-time research assistant. The job involves keeping an entire research operation moving. Lab managers oversee recruitment needs, train and communicate with research assistants, handle lab finances, and troubleshoot all kinds of unexpected problems. Many lab managers get to do research on top of all of this, but communicate how you can work independently and think on your feet.
Treat all of your research experience as relevant.
You do not need to have worked on exactly the type of research the lab conducts. What matters is that you can explain what you did, so they can see how you can be useful to them. When describing your research experience, be specific about your responsibilities. Did you recruit, schedule, and run participants? Did you train research assistants? Did you write IRB protocols or submit amendments?
Emphasize special responsibilities.
Take ownership of the times you have gone beyond simply following instructions. If you were a research assistant in the lab for several years, did you train a new research assistant on a task? Have you ever independently designed a study and managed recruitment for it?
Be specific about why you want this position.
Have a clear reason for doing a post-bacc and for wanting to work in this lab. Being excited to do scientific research in this lab can set you apart from the other applicants. Look at the PI's recent grants, because they are probably what will be funding you. Do you have any ideas that connect to it? How about any of their recent articles where they are first or last author (if they are middle authors, the work may be more tangential for them).