Use Fetchmail on the link between your provider and your email server. Schedule a fetch every 5 minutes or so.. This makes for a very flexible and resilient system.

My mailboxes are on a Dovecot (IMAP) server.

On DNS there is an MX record pointing to the server.

To read the mail I use Roundcube.

If the server is down, I lose the new mail.

I would like to do two things, alternatively.

First option.

1) I keeep the mail on the provider's server, restoring the original MX record.

2) Dovecot does not receive the mail, but should go to download it.

3) I can access the server with Roundcube to read the downloaded mail, but if the server is down I can always go to read the mail on the provider's server with its webmail

Second option

I install a second mail server and the two have to replicate the mail.

But I don't know how to do it, so I need a guide with instructions.

Furthermore Roundcube would always access only one server and if this is down, how can I connect to the second one?

This is a typical situation for webmail like Yahoo, Outlook or GMail, but I'm not Yahoo, Microsoft and Google ;)

Thanks in advance for any advice.