Thats not really how it works. Lemmy works on a Push model. When a post is created and the instance knows you are interested in the post by a user subscribing to it, the Lemmy instance will send the post to your server. But if your sever does not respond then the other instance will mark your server as dead and stop trying to send you posts.
When you turn on your instance you aren’t going to get a flood of posts that you have missed over the past few days/hours.