I think you have it backward. Notifications come from your email client after it polls the email server. So you need an email client on your android device that checks the server.
No need to self-host anything, unless you want to self-host your own email server, something that (while doable) is NOT to be taken lightly and will undounbtely lead to difficulties and lots of hiccups down the road.
I am currently self-hosting my email server (bee, doing for almost 20y) but not at home and absolutely can confirm it's very complex to setup properly.
Up and down. Actually lots of things are falling into place, and a few "end points reached" are proving to be... just new starting points.
What to say, need to get my knee in order tough. After last marathon it is still giving me issues and i am fed up to wait and want to go back running....