The best way to reach Ein Karem if you are coming from out of town, is to catch the Jerusalem lite train at the Jerusalem central station (or from Jaffa st. if you arrive by Nesher shuttles) and get off at the last stop (about 10 mins. ride) at mount Herzl.
From there you will catch bus # 28 that comes all day long every 15-20 mins and get off at the first stop in Ein Karem (about 2 mins. ride) which is right next to the house. Nice and very easy!