I just took this bus service today, from Dubai to Abu Dhabi.
The Dubai bus station is across the street from the Al Ghubaiba metro station. If you are arriving at DXB at a reasonable hour (I'd say before 9p), you can take the metro. Take the Red Line toward Jebel Ali, transferring at Khalid bin al Waleed to the Green Line toward Etislat. Get off at Al Ghubaiba and take Exit 1. Take the crosswalks just past the gas station across the street to the bus station.
There is a ticket counter on the far side of the station where you can buy your ticket. Not sure if credit cards are accepted or not, but the total cost is only 25AED (opposed to a 300AED taxi ride). It will drop you off in the middle of Abu Dhabi. You'll need a taxi from there to get to AUH, which should be under 70AED.
I do think the thread title should be edited to reflect that this bus service is NOT between airports - it's between city centers. Will put in a request for that!