When installing a service pack on one instance will this hinder the running of the other instances on a standalone SQL Server?
The Service Pack installation process will identify if there are any files in use that will necessitate a restart of the server.

Each instance has it's own independent set of executables, so instance B should not be affected by installation of a new service pack on instance A.

The SQL Server Browser Service will be restarted pretty quickly, however there may be a short period of time where name resolution for named instances may not work as expected.

Best practice for this kind of work on a production server is to schedule it to occur during a period of low or no activity.