You could create a IMAP account for fastmail using the free version of the em client from
http://www.emclient.com/ . Menu -> File -> Import appears to support importing mail, calendars and contacts using the file formats you are using. The main limitation of the free version is its limited to 2 accounts.
Importing the .eml files will make the messages available under webmail (since IMAP accounts support remote folders, stored on the mail server). According to
https://www.emclient.com/faq-account-setup it supports CardDAV (remote contacts) and CalDAV (remote calendar) so you should be able to synchronize your address book and calendar (unless you have a legacy plan that doesn't support CardDAV and CalDAV).