from your description it looks as if a smart collection would do the job, but there you cannot define a rule based on a date.
So you need to define a tag 'new arrival' and manage this tag on the affected products. You can do this tag management manually or via an app like "Simple Bulk Tag Editor", see https://apps.shopify.com/simple-bulk-tag-editor-scheduler (no connection to us/never used it). We would do it with a scheduled python script with requests to the Shopify Admin Rest API.
The New Arrivals Shopify app will perfectly fit your need.
It can automatically tag and untag products based on creation or publishing date. You can set up a concrete period of time the products should be considered to be "new". It will automatically remove the tag after a specified period of time.