You'll need to use MimeFactory to create mime entity from the appointment object. Then simply use MailBuilder to create email message and add mime entity as a attachment:
Appointment appointment = new Appointment();
Event e = appointment.AddEvent();
e.Description = "Status meeting description";
e.Summary = "Status meeting summary";
e.Start = new DateTime(2015, 05, 10, 16, 00, 00);
e.End = new DateTime(2015, 05, 10, 17, 00, 00);
e.SetOrganizer(new Person("Alice", "alice@mail.com"));
e.AddParticipant(new Participant(
"Bob", "bob@mail.com", ParticipationRole.Required, true));
e.AddParticipant(new Participant(
"Tom", "tom@mail.com", ParticipationRole.Optional, true));
e.AddParticipant(new Participant(
"Alice", "alice@mail.com", ParticipationRole.Required, true));
Alarm alarm = e.AddAlarm();
alarm.BeforeStart(TimeSpan.FromMinutes(15));
MailBuilder builder = new MailBuilder();
builder.Text = "text";
builder.From.Add(new MailBox("alice@mail.com", "Alice"));
builder.To.Add(new MailBox("bob@mail.com", "Bob"));
MimeCalendar att = new MimeFactory().CreateMimeCalendar(appointment);
builder.AddAttachment(att);
IMail email = builder.Create();
using (Smtp smtp = new Smtp())
{
smtp.Connect("smtp.server.com"); // or ConnectSSL for SSL
smtp.UseBestLogin("user", "password");
smtp.SendMessage(email);
smtp.Close();
}