How would I do $push using MongoRepository in Spring Data?

Yes, you must implement a custom method on the repository and your push method would be something like this :

public class FooRepositoryImpl implements
    AppointmentWarehouseRepositoryCustom {

    @Autowired
    protected MongoTemplate mongoTemplate;

    public void pushMethod(String objectId, Object... events) {
        mongoTemplate.updateFirst(
            Query.query(Criteria.where("id").is(objectId)), 
            new Update().pushAll("events", events), Foo.class);
    }
}

You can do this but I ran into an issue where the "_class" field wasn't being preserved. 

The pushed object itself was run through the configured converter but for some reason the "_class" field of that pushed object wasn't written. 

However, if I injected the converter and wrote the object to a DBObject myself, then the "_class" field was preserved and written. The thus becomes:

public class FooRepositoryImpl implements
AppointmentWarehouseRepositoryCustom {

@Autowired
protected MongoTemplate mongoTemplate;

public void pushMethod(String objectId, Object event) {
    DBObject eventObj = new BasicDBObject();
    converter.write(event, eventObj);
    mongoTemplate.updateFirst(
        Query.query(Criteria.where("id").is(objectId)), 
        new Update().push("events", eventObj), Foo.class);
  }
}