MongoClientSettings settings = new MongoClientSettings();
settings.ReplicaSetName = "MyReplica";
settings.ConnectionMode = ConnectionMode.ReplicaSet;
settings.ReadPreference = new ReadPreference(ReadPreferenceMode.Secondary);
settings.Server = new MongoServerAddress("Host2", 27017);
MongoClient client = new MongoClient(settings);
var dbadmin = client.GetDatabase("admin");
//Класс ReplicaInfo соответствует JSON ответу на rs.Status()
var replSetGetStatus = dbadmin.RunCommand<ReplicaInfo>("{ replSetGetStatus: 1 }");
//Класс Member соответствует описанию члена реплики по rs.Status()
Member memberInfo = replSetGetStatus.members.First(x => x.Self);
Console.WriteLine(memberInfo.Name + ":" + memberInfo.StateStr);

[BsonIgnoreExtraElements]
class Members
{
    [BsonElement("self")]
    public bool Self { get; set; }
    [BsonElement("name")]
    public string Name { get; set; }
    [BsonElement("stateStr")]
    public string StateStr { get; set; }
}

[BsonIgnoreExtraElements]
class ReplicaInfo {
    [BsonElement("members")]
    public List<Members> members { get; set; }
}