Email is all based on online servers maintained by various services (Google, Yahoo, AOL, or some indie running their own email server) so it's always a server <> client situation. Any email app is always going to be the client so it doesn't have the ability to do things like switch from POP3 to IMAP.
You need to log into your email service account and use the web interface to do the changeover.

Which email service are you using?
If you haven't been allowing your email service to save everything for you (so there are copies in your online account),it's not safe to rely on your phone as the only source for your data. You really need to get a backup solution set up now. Try setting up Samsung's Smart Switch utility:
https://www.samsung.com/us/smart-switch/

 

Yes, this 'should' work. But it might depend on scale, approximately how many email messages are there? A few dozen, a few thousand, or more? Also, do a lot of those old emails have attachments?
https://support.google.com/mail/answer/21289?co=GENIE.Platform=Desktop&hl=en

 

The only way I can think of is to create a new email account with imap, then login to your old email account and send those emails to your new imap email account....