No, Spring does not use AOP in order to find the correct handler for the requested URI.
That's essentially the job of the DispatcherServlet.
@GetMapping, @PostMapping, @PutMapping, @PatchMapping, @DeleteMapping are all shortcuts for their respectives @RequestMapping annotations.
The DispatcherServlet is a front controller that handles the incoming requests and delegates them to the correct handler.
When a request arrives, the DispatcherServlet uses HandlerMappings and HandlerAdapters from the WebApplicationContext to delegate the request to the appropriate handler. Of course it's a lot more complicated then just that and there's a lot of engineering going on.
The DispatcherServlet doesn't invoke the handler method directly, instead, the HandlerAdapter interface is used. Specifically, in the case of @RequestMapping annotations and it's shortcuts, the RequestMappingHandlerAdapter is used together with the RequestMappingHandlerMapping.
The RequestMappingHandlerMapping is used by the DispatcherServlet to obtain the handler object (when using @RequestMapping in a class instead of a method) and the handler method (when using @RequestMapping and it's shortcuts in a method). This class is responsible for creating the RequestMappingInfo for each @RequestMapping annotated class or method, so, basically, it maps the corresponding handlers to it's paths.
The RequestMappingHandlerAdapter is used to invoke the handler method through the RequestMappingHandlerAdapter#invokeHandlerMethod (see here)
So no, Spring doesn't use AOP in this case. It's a lot of intelligent mapping going on.