You're probably going to have to switch to using either IMAP or EWS to talk to your exchange box.
Also, I found this snippet:
All MAPI traffic is RPC-based. Historically, Outlook clients have had
two transport methods for RPC traffic available to them: RPC over TCP,
and RPC over HTTP (aka Outlook Anywhere). In Exchange 2013, we removed
the RPC over TCP option, leaving RPC over HTTP as the only
connectivity method. This does not mean RPC is no longer supported. In
fact, it is still used (e.g., we still make MAPI/RPC calls), we just
encapsulate them in HTTP packets.
So, it stands to reason that your old MAPI-based code could still work if made to use HTTP. Not sure what that would entail, but if it saves you the trouble of converting to a different protocol (read: server config instead of code changes), it could be worth the research time.
One last note: take a look at this MSDN page that discusses MAPI over HTTP.