Java has a happens before/happens after memory model. There needs to be some common concurrent construct (e.g. synchronized block/method, lock, volatile, atomic) on both the write path and the read path to trigger this behaviour.
If you synchronize both methods you are creating a lock on the whole object that will be shared by both the read and write threads. The JVM will ensure that any changes that occur on the writing thread that occur before leaving the (synchronized) setInt method will be visible to any reading threads after they enter the (synchronized) getInt method. The JVM will insert the necessary memory barriers/cache flushes to ensure that this will happen.
If only the write method is synchronized then changes to the object may not be visible to any reading thread. This is because there is no point on the read path that the JVM can use to ensure that the reading thread's visible memory (cache's etc.) are in line with the writing thread. Make the getInt method synchronized would provide that.
Note: specifically in this case making the field 'number' volatile would give the correct behaviour as volatile read/write also provides the same memory visibility behaviour in the JVM and the action inside of the setInt method is only an assignment.