Microsoft's official position is that you cannot access Shared Mailboxes in an Office 365 tenancy from an Android device. Here's how to do it by configuring a new IMAP account with the following settings:
Incoming Settings
1. Email Address = SharedMailbox@Domain.com
2. Username = O365Account@Domain.com\SharedMailbox
3. Password = Password
4. IMAP Server = outlook.office365.com
5. Security Type = SSL
6. Port = 993
7. IMAP path prefix = Optional
Outgoing Settings
1. SMTP Server = smtp.office365.com
2. Security Type = TLS
3. Port = 587
4. Require Sign In = Yes
5. Username = O365Account@Domain.com
6. Password = Password
You should be able to have this work also on iOS and Android (and maybe even Mac OSX) devices by setting an account up in a similar fashion.
Regards,
The Outspoken Wookie