The easiest way is indeed a taxi, and yes, that price sounds about right, although it's a regular metered fare so the exact cost will vary. Not every taxi will be willing to up and off to Abu Dhabi on a lark though, but the airport taxi stand can sort one out if you ask, or you can call and book a cab. As for safe, well... most UAE cabbies drive fairly sanely by the standards of the region, but the Dubai-Abu Dhabi highway always has the occasional blinged-up rich kid doing 300km/h in the fast lane in Daddy's Lamborghini.
The cheaper and safer, but slower and less convenient, option is the Emirates Express bus, Dhs 20 one-way. Buses leave from Ghubaiba station, on the Dubai Metro Green Line, every 30 min or so.