For a character who only shows up at the beginning and ending of the story, King Arthur sure gets a lot of time devoted to describing him. As soon as the Christmas feasting begins, we learn that Arthur is too excited to eat, for "he was so lively in his youth, and a little boyish" (86). In fact, Arthur has a hard time sitting still, he’s so restless.
He also refuses to eat on a feast day until he’s witnessed something marvelous: like Lord Bertilak, amusement and merriment seem to be a huge priority for him. In that, though, he’s just being a good host, for it’s his job as lord of the castle to make sure that all his guests have a good time at his party. For the same reason, Arthur works hard to appear unphased after the Green Knight rides away carrying his own head, assuring Guinevere that "such strange goings-on are fitting at Christmas" (471).
Arthur also seems to be extremely brave: after all, he’s the first to volunteer to take a swing at the Green Knight and, in fact, the poem describes him as "by nature bold" (321).
One of the reasons the poem takes time to characterize Arthur despite his relatively minor role in the plot might be that this is a character with a history that extends far beyond this single poem. In early medieval romances, the character of Arthur comes to stand for an ideal king, his court and knights, a system of chivalry relatively un-plagued by the corruption that haunted real-life medieval kings and courts. And although that characterization was to change later on, Sir Gawain and the Green Knight represents Arthur as a great king and his court as a happy, civilized refuge for knights-errant like Gawain.