Will one SQL instance work whilst the other 2 are patched?
Firstly, it always better to test patches in DEV environment before applying on Production. 

To answer, there won't be downtime/interruption for Instance 3 while patching/removing path on Instance 1, all you need select particular instance during the wizard or command line as follows:

KBXXXX.exe /qs /IAcceptSQLServerLicenseTerms /Action=Patch /InstanceName=Intance1


There might be restart required (manual restart) to proceed with Instance 2 patching after completing instance 1.

Shared features are updated (patched) once. In your case during the Instance 1 or which ever starts first, and not applied while patching other instances on the same server. We never experienced any interruption with shared features (considered following): But if you got Integration Services and it's heavily used you must find maintenance windows where it's got less overhead. 


Client Tools Connectivity
Client Tools Backward Compatibility 


My environment was 3 instances on 1 server, all of them are AG replicas. One primary other two secondary, and there are 3 servers total with same configuration.