Unfortunately, Gmail filters (rules) only applies on incoming mails, not mails already in the inbox. You can still do a search and manually clean up your inbox by using a very specific search.
Let's say that all your emails you don't want to keep forever are labeled Newsletters.
To display all emails labeled Newsletters that are older than 2 months, use the following search query
You can replace the older_than parameters to something else like 7d
- d: days
- m: months
- y: years
I guess a Google Apps Script could be able to take care of this, but I don't have experience (yet) on that matter.