I used the OpenWrt 22.03.5 image downloaded from the archive and after executing the image I am able to find the directory /usr/lib/lua/controller/admin.
When I compile the OpenWrt from source code and after running the compiled image, I couldn't find the /usr/lib/lua/controller/admin directory.
Do anyone know what is going on? Is there anything I should enable or do when I compile the source code?
Usually if you use a developement build, luci needs to be installed manually.
When you compile OpenWrt you need to include luci in your build otherwise it will not be installed by default, it will look more like it is a dev build, you should be able to access it via make menuconfig
Or did you do already this step?
-edit-
Aha, when I was looking myself I see the path does also not exists but I think the controller is moved to /usr/lib/lua/luci/ I'm not so aware of the luci api, but it looks like it is in there.
I can see the directory /usr/lib but not inside the lib directory there is no lua directory. Is there any selection I need to select in the menuconfig?
There should be a option for luci, as parent option/node in make menuconfig if you open the tree you will have multiple options for example: luci-ssl if you select the normal luci there it should work I think.
if the folder still misses then I'm not sure why this is, I hope someone else can assist you with more experience 
Have you selected LuCi during compilation time at ‘make menuconfig’ phase?
Try to issue ‘make menuconfig’ and choose LuCi packages, after that you should be able to see LuCi files
By the way; what is the model of the router you use?