HIRE CAR

If you hire a car the distance is 162 km and the journey would take about 1 hour 55 mins.

There is  good scenery. You have to drive through the desert. This is where many of the Spaghetti westerns were made such as The Good the Bad and the Ugly, For a Few Dollars More etc.  The striking mountains in the distance provide the perfect backdrop.

Driving instructions are here.
http://crazycalculations.com/directions/index.php?to=Granada&from=Almeria

IDEA:
You could have a very interesting holiday by hiring a car spending a few days on the very beautiful Cabo de Gata natural park area to the east of Almeria which has a fairly unspoilt coastline with no high-rise hotels and good beaches.
Then also spend some days in Granada and maybe go on excursions to the Alpujarras mountains.

BUS
The fastest bus takes about 2 hours, the price is about 14 euros, there are 8 buses per day.
See more info here
http://www.alsa.es

SHUTTLE SERVICE
Acording to the price calculator a shuttle taxi would cost 88 euros per person.
So probably too expensive for most people.
http://www.shuttledirect.com