As I can see, Update case is a quick action which is added to the layout in Salesforce Mobile and Lightning Actions section.
A quick action can be of different type but in your case, it seems of update a record type.
As it is coming in Feed tab, You have enabled feed tracking on case
object. If you disable the feed tracking for the case object, It will come to top right-hand side button
list on record detail page.
Salesforce clearly mention this point in there documentation:-
Actions on user profiles, cases, and work orders can appear in a
different way than on other records.
When feed tracking is enabled for cases or work orders, the page-level
action menu on those records contains only custom buttons and
supported standard buttons. Quick actions appear on the Chatter tab.
Reference:- Actions in Lightning Experience
Update:- Actions are of two types in salesforce.
- Object-Specific Actions:- Object-specific actions let users quickly create or update records, log calls, send emails, and more, in the context of a particular object.
step to finding object specific action:->>> setup-->> Object manager-->> select the object-->> button links and action
- Global Quick Actions:- You can add global quick actions to almost any page that supports actions. Use global actions to let users log call details, create or update records, or send email, all without leaving the page they’re on. Global create actions enable users to create object records, but the new record has no direct relationship with other records. e.g:- Send Email, Log a Call etc.
step to find all the global action:-
- From Setup, enter Actions in the Quick Find box, then select Global
Read more about them here:-
- Object-Specific Actions
- Global Quick Actions
So in your case, If it is not available in object's Buttons, Links,
and Actions, then check for global action from the setup menu.