Entry to Saudi Arabia via causeway

What you heard is only partly true.

For Saudi Arabia, every expat (irrespective of residency or profession on visa) needs to apply for and get a visa in advance.  And a company in Saudi needs to sponsor you.  The only un-sponsored visa available to most, is the business visit (investor) visa which you would not be eligible for.  They are supposed to be launching a tourist visa soon so let's see when that happens.  The application process is through VFS Tahseel in Bahrain Financial Harbor and you can browse their website.   Oh and by the way, the cost for most nationalities for any kind of visa is very high; prepare to pay a minimum of 200 BD for a single visit visa for Pakistanis and around 350 or so for 6 month multiple entry.

Similarly, for UAE, you need to apply in advance for a visa online as a GCC resident if you belong to a nationality which doesn't get visa on arrival to UAE. For example, a Pakistani with Bahrain residency would need to apply for a visa to enter UAE but a British resident would not as he can get visa on arrival due to his passport i.e. he will enter as a British national not GCC resident.

In the rest of the GCC countries, you can get visa on arrival.  Although I am not sure how long that will last as most of them are debuting e-visa systems in parallel i.e.  you have the choice of either e-visa or on arrival visa.  UAE started this way and then made e-visa the only option for GCC residents.