They used to provide you with a '3G' stick upon arrival with a limited amount of data, Im not sure if they still do.
It usually takes a few weeks to get your home internet set up, and you cant purchase outside internet until you get your iqama (Which again takes a few weeks).
You usually get your login ID within a few days and will have access in the library and wifi areas (which depend on your camp).
Sometimes the best bet is to simply meet your neighbors and ask if you can use theirs. All of us were new at some point, those who've only been here a few years remember what it was like to start here with nothing and many are willing to help out.
You can get a temporary wifi hotspot from the STC or other mobile shops in Dhahran. Very convenient if you get STC because it is at the Mujama building which is right off the core area where buses arrive from other camps.
This should tide you over till Aramco gets you their connection. I got mine on the second day here. Very tough to manage business affairs at home without it since so much is being done electronically these days.