While catw sometimes contributes valuable information for tourists to this forum, in this case he/she is terribly misleading you.
I will repeat once again - The drive east of Jerusalem on Hwy 1 and south on Hwy 90 does take you through some territory that Israel conquered from Jordan in the 1967 Six Day War. These roads are NOT, however, located in territories administered by the Palestinians. You will be driving near Jericho and can see the city off to the left of Hwy 1, but the road bypasses that Palestinian city entirely.
There is a single checkpoint on the journey from Jerusalem to Masada will be on Hwy 90 going south. I have accompanied tourists on that journey countless times. Never has our car been stopped and never have the tourists been asked to present a passport. Even if they were asked to do so, the Israeli authorities will NOT be viewing this as entering or exiting Israel. On the way back to Jerusalem, there is a single checkpoint on Hwy 1 near Ma'ale Adumim. I have never seen tourists being asked to produce passports there either and even if it happens, the authorities will never consider that a "re-entry" of the tourist into Israel.