To whom is the Domain Name registered? I`m surprised you run a business based on using Web Mail. You should set up the e-mails to download to your system and use webmail for remote access only.
Contact your "guy" and request a detailed description of the service he is providing and associated costs, including any service element and contract periods, which might be a monthly fee. Also request he specify any limitations to the service and request user names and passwords for e-mail access, ftp access and Control Panel access. (If anything happens to him you may be in even further trouble)He should be able to provide this information within a few hours.
Then you can take some action by formally writing and expressing your concerns and consider moving provider.