First, it is not possible for two invocations of synchronized methods on the same object to interleave. When one thread is executing a synchronized method for an object, all other threads that invoke synchronized methods for the same object block (suspend execution) until the first thread is done with the object.

Can we synchronize a method?

Yes, we can synchronize a run() method in Java, but it is not required because this method has been executed by a single thread only.

When should a method be synchronized?

We need to synchronize the shared resources to ensure that at a time only one thread is able to access the shared resource. If an Object is shared by multiple threads then there is need of synchronization in order to avoid the Object's state to be getting corrupted. Synchronization is needed when Object is mutable.