Exim is only a "mail transport agent" (i.e., SMTP server); this will allow your machine to send and receive mail from other mail servers, but in order to make your mail readable by a client (webmail, outlook, etc.) you need a mail delivery agent (a.k.a. POP/IMAP server) like courier or dovecot, which handles user inboxes. The Linode email guides are pretty straightforward to get these set up.
In order to receive mail for a domain on your Linode, you need to have an MX record for that domain pointing to your Linode. You should be able to configure Exim to do domain forwarding (
user@domain2.com =>
user@domain.com).