It's important to note that Al-Saud (the way his name was listed in his NASA bio, now gone down the memory hole like most of jsc.nasa.gov) paid for his ticket.
Back in those days of flying commercial payloads on the Shuttle, it was possible to pay for a payload specialist seat as part of "Standard Launch Services".
(emphasis mine, scanned from STS Customer Accomodations JSC-21000-HBK, not currently online, in the queue to be scanned and uploaded)
See also Rodolfo Neri Vela of Mexico who flew as a payload specialist when the Mexican communications satellite MORELOS-B was launched from STS-61B.
Their countries paid for the payload, which included the payload specialist seat, and got to pick who flew.
What they actually did on the flight was of secondary importance. Unfortunately, these missions flew too long ago to have their flight plans posted online, which would be the definite answer of what each crewperson did.
Note that Spacefacts states
Sultan Salman Al-Saud, the Saudi Arabian payload specialist was not
involved in the deployment activities of Arabsat.
but goes on to say
Sultan Salman Al-Saud, the Saudi Arabian Arabsat payload specialist,
took part in four scientific investigations: Earth Observation, Phase
Separation, Ionized Gas and French Postural Experiments. Salman
Al-Saud photographed Saudi Arabia during Discovery's daylight passes
over its southwestern region with a 70 mm camera from orbit. The
photographs were be studied by Saudi scientists at the research
institute, the University of Petroleum and Minerals in Dhahran, Saudi
Arabia. They also were compared with previous data from multispectral
scanner, thematic mapper and radar images (SIR-A and SIR-B). Analysis
covered geological features, sand dune morphology, hydrogeological
features, turbidity in the Red Sea, urban areas and forestry.
another experiment, two liquids which do not mix on Earth were studied
in microgravity. They were referred to as "phases." Using Phase
Separation Experiment hardware developed at the NASA Marshall Space
Flight Center, Salman Al-Saud placed various concentrations of Saudi,
Kuwaiti and Algerian oils mixed with water in a hand-held, transparent
Plexiglas container with 15 chambers, each having a small metal mixing
ball. He shaked the container and mount it in front of a fluorescent
light, then photograph the separation and record his observations [sic].
Ionized Gas Experiment (ICE) required that Sultan Salman Al-Saud,
using the Orbiter Discovery's TV cameras, recorded thruster firings in
specified configurations to study the mechanics of thruster plumes and
the degree off Ionization produced. This experiment was part of a
Saudi Arabian student Ph.D. thesis at Stanford University. Purpose was
to assess the thruster plumes effect on operations measurements and
communications associated with space vehicles. The data obtained also
were analyzed at the university of petroleum and Minerals for a better
understanding of the impact of gas particles on solid surfaces. Salman
Al-Saud also photographed the Arabsat satellite's rocket engine
Salman Al-Saud assisted French payload specialist Patrick
Baudry in the postural Experiment on the adaptation mechanism of the
sensory motor activities. This included posture stabilization and
orientation and the role of vision in posture control and reflex
mechanisms that stabilize the retina. Requiring about 3 hours, the
experiment was performed before, at the beginning, during the middle
and last day of flight two times a day.