firstly, 4 days is WAY too short for what you want to do. i have been to israel proper and the west bank on two different trips, most recently in March. you can enter the occupied territories with a EU passport; well, let me clarify that, you can enter the west bank, but not gaza. you can enter from israel proper; i always entered the west bank from east jerusalem, but there are other entry points. i cannot be 100% certain whether they will allow you to drive a rental car in there. if you rent a car in israel proper, it will have yellow israeli license plates. with those license plates, you may only be able to drive on "bypass roads" in the west bank, i'm not 100% certain. but why do you want to rent a car? public transport is so good in israel proper, and even in the west bank, it is sufficient.
if i only had four days in israel, my #1 destination would be Jerusalem, with at least one FULL day in the Old City, but preferably two days; there is MUCH to see in the Old City. i think the west bank is interesting because it's a contrast to israel proper. however, your time is limited and the checkpoints in the west bank can take time to cross, especially if for some reason, israel closes one. are you a spiritual pilgrim? if so, i would probably take one day for bethlehem and hebron; go to bethlehem in the morning and hebron in the afternoon. take public transport from outside damascus gate of the old city in the west bank; may be easier than driving a rental car in. bethlehem and hebron are close; but do not go on Shabbat (Saturday) or you will get stuck trying to get back to jerusalem. if i were you, i would sleep in jerusalem all nights. israel is small enough to do this. i usually based myself in jerusalem and made day trips to different places in the west bank (bethlehem, hebron, jericho, ramallah, nablus, ni'lin, jenin, leaving early in the morning no later than 8am or so). from jerusalem a very nice day trip is masada and the dead sea. i would recommend this over the cesarea ruins. the sea of galilee area is nice for it's biblical history. go alone, taking a bus, or drive to Tiberias, then visit the Sea and churches, etc., from Tiberias, or take a one-day tour from Jerusalem as i did in 2004. use United Tours, i think it is $70 US Dollars or so for one day; it is informative and they go to many sites; pick you up in jerusalem, or can leave from tel aviv.
four days is not much, my dear. but jerusalem is a MUST!!!!!! DO NOT MISS JERUSALEM, ESPECIALLY THE OLD CITY!!!!! besides jerusalem, you will have to decide maybe one day in West Bank and maybe one day in Sea of Galilee region. Akka is okay, but can be missed if you only have 4 days. if i were you, 2 days in Jerusalem, one day in Bethlehem and Hebron (in Hebron, visit the Old City and the Tomb of Abraham). the other day, either Dead Sea and Masada or take a day-trip tour on "United tours" to the sea of galilee region. don't rent a car, it's a waste of money and u may not be able to drive it into the west bank.
IM me for more details if u want.
- julie