The version of SSMS is wholly irrelevant here - the error is referring to SQL Server itself.
In this case you will need to upgrade your SQL Server from 2012 to 2014.
How can I upgrade SQL Server from 2012 to 2014?
As this is Express edition and likely not a production server, I would recommend an inplace upgrade. You will need to download the installation media for SQL 2014 Express here.
Once downloaded, run it and choose the option to Upgrade from a previous version of SQL Server and follow the Wizard from there.
I would advise to take full backups of any databases you can't lose before doing this.
Are there any other ways to do this with less downtime?
You could also perform a side-by-side install of 2014 with 2012, and then migrate the databases over by backing them up then restoring them.
Please note this could impact how people connect to this instance - you can't create the new instance with the same name as the old instance, so if this is how users are connecting you may need to set up new connection strings.
As an aside the version of SQL 2014 you are running has several known issues and security problems - it's 2014 SP1. There have been 2 SP's released since then - I would recommend you speak to whoever is in charge of this box and look to get SQL server upgraded to the latest SP if at all possible.