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I'm a student and it seems every school or university I have been to has one password that you set for your user account for logging in to university services, which is also then synced to external services the university use such as blackboard, fronter, dropbox, Office 365 e-mail, etc.

Lesson 1 of cybersecurity is not to store passwords in plain text or to encrypt them. But instead to use some sort of hashing algorithm. If this is true how can a university's IT service automatically sync password for all the relevant accounts? I can understand how this can be done by using APIs each time to update all the services when a password change has been requested, but it would then make it impossible to adopt a new service without the user re-entering their password.

How is it done? or are they just holding passwords in plain text?

It's usually not that passwords are "synced" between services, but rather a centralized authentication service is used. In many cases, this is going to be a Microsoft Windows domain controller running an Active Directory server (others exist e.g. FreeIPA), which other services can talk to using LDAP and Kerberos.

The typical setup has all user accounts added to the directory server (which is usually replicated across multiple servers transparently for redundancy and reliability purposes). Locally hosted applications (e.g. Blackboard) will have the directory server's LDAP information entered into the server settings as an authentication provider. When a client enters their credentials on the web interface, the application may check the credentials against a local database as well as LDAP services that have been configured. If the LDAP server confirms a successful authentication, information about the user (contact info, group membership, etc.) can be retrieved to populate parts of the application. When user information is updated somewhere, the data on the directory server is changed so that the change will be visible everywhere else. This applies to changing the password.

Not all applications will use LDAP directly; external services such as Office 365 or Google Apps suite and others may instead use single sign-on (SSO), where you authenticate through your organization's login page and these external services are able to effectively reuse this authentication (e.g. through SAML).

On the directory server, passwords are stored as hashes within each user object. The hashes are protected further using LDAP access controls (so any LDAP client can't just pull hashes) and are encrypted with a key from the registry (in the case of Windows Active directory).

In short, this is just scratching the surface. There are a number of ways to set up a network and services to use centralized authentication. But almost every organization uses some type of this system; like you said, it would be very difficult and insecure to do manually in most cases.