If you see a job you want, apply for it as quickly as you can make a good application.
I would expect anyone who is actively looking for a job to be regularly checking job posting sites and prepared to apply on short notice when they see one they are interested in. This does not come across as "desperate".
The order applications come in often doesn't matter at all. Depending on the institution, the people making the decisions might not even know this (applications may go through an HR department and be handed over to those hiring in a single pack).
For applications through this site, it seems that the date of application can be easily seen. But it is still doubtful whether anyone will notice or care. To the extent that it is noticed and signals anything, I would say it is a positive signal. It suggests:
- You have a high level of interest in this job.
- You are motivated.
- You are organized and prepared.
All of these are positives from a hiring perspective.
More important than timing is putting in a strong application. If you see a job early, it might be wise to spend extra time researching the organization, preparing your application, and tailoring your CV to focus on the right things. But assuming you are fully prepared, there is no downside to applying on the first day.