What’s your existing setup? For such a simple task, check if any of the tools you use currently can be adapted (simple text files on a web server? File sharing like Nextcloud and text files? Pastebin-like? Wiki? …). Otherwise a simple Shaarli instance could do the trick (just post “notes” aka. bookmarks without an URL). I use this theme to make it nicer. Or maybe a static site generator/blog.
dovecot will be able to handle this part. This is what I use as a mail archive (once a year, archive all mail from the previous year from various mailboxes to my self-hosted dovecot instance). I wrote this ansible role for it.
As others recommended,
imapsync
should be able to handle that part.These tools are simple enough to install and manage (one package, one config file), Docker is not needed. If you really need it to fit into your docker-based setup, build and maintain your own images.