Why did the Predator not kill Arnold immediately as it killed everybody else?

As shown throughout the film, the Predator has only killed armed opponents. At this point in the film Arnold has put up a great fight against the Predator and is currently unarmed. The Predator looks him over much like a hunter might take time looking at a prize kill before firing a shot. The Predator disarms himself and then fights Arnold hand-to-hand, a "fair" fight instead of just killing defenseless prey. (It's been a long time since I've seen the film, so can't remember exactly when/how the Predator stops using weapons).

A couple of other reasons (besides the already mentioned one: Dutch was unarmed at the time so the Predator chose to disarm himself and to engage in melee combat):

• Dutch was a fantastic adversary; he was clearly the strongest, the smartest, and he lasted the longest among all the preys. The delay in killing was in part to show some sense of respect.
• Dutch was also the last of the group: the Predator was probably milking the hunt because if it killed Dutch, it wouldn't have another foe (for a while at least). In combination with the first point above: the Predator knew it wouldn't get such a great foe like Dutch again, possibly forever.

As another line of reasoning:

The predator is sometimes mocked as the dentist-on-safari among alien invaders. With all previous kills it had more opponents. So it killed them off instantly and with all its high-tech gadgets to avoid the risk of being outnumbered/caught off-guard.

Dutch (Arnold's role) was alone in the end. So the dentist-on-safari Predator tried to prove to itself what great a hunter he is. And did it go well...?

Man, Predators really suck, if looking at them like this...

You mean this scene?

The Predator collects the skulls of his prey as trophies, so he was probably checking if Arnold's skull was worthy of his collection.

After that he removes his helmet and goes without weapons, to make this hunt more challenging since he would be lowering himself to the technical level of his prey.

The others were way too easy to kill, so perhaps not as good as a trophy.

Answering the last part of your question: "Did checking Arnold's head mean something?"

I always thought that the predator examined Arnold to try to understand the reason why he wasn't able to see him (the mud).

Arnold still had sections of his face covered in mud so the Predator "wonders" how these sections are not visible to him and others are.