In Beowulf, how might Beowulf's virtues be especially important to a warrior society?
Beowulf is brave, a fierce warrior, and he's proud of his accomplishments. Whether this is pride or confidence, Beowulf's personality and physical abilities make him the ideal warrior. Beowulf also has great appreciation for the legacy of his ancestry and his own future legacy. There is a fine line between Beowulf's selfish boasting and his well earned status as a great warrior. In this sense, Beowulf has earned the right to boast. So, even when he does brag about his victories, he is simply describing his exploits. When Unferth challenges Beowulf, saying he lost the swimming competition to Breca, Beowulf counters by saying that he would have won if he didn't have to battle a sea monster. He then criticizes Unferth, saying if he (Unferth) was a decent warrior, Hrothgar would not have needed Beowulf to come and take care of Grendel:
I tell you truly, son of Ecglaf, that Grendel, awful monster, would never have performed so many terrible deeds against your chief, humiliation in Heorot, if your spirit, your heart, were so fierce in fight as you claim. (IX)
Beowulf will not back down from a challenge and he's willing to prove his worth by defeating Grendel. For Beowulf, praise is meaningless unless it is backed up by action. That's why he criticizes Unferth because Unferth has done nothing to defeat Grendel himself.
What also makes him the ideal warrior is that he isn't one to rest on his laurels. He continues to seek out challenges even well into old age. Years later, he is even willing to fight the dragon alone:
Wait on the barrow, safe in your mail-shirts, men in armor--which of us two may better bear wounds after our bloody meeting. This is not your venture, nor is it right for any man except me alone that he should spend his strength against the monster, do this man's deed. By my courage, I shall get gold, or war will take your king, dire life-evil. (XXXV)
Here again, Beowulf is the ideal warrior, being proud, still seeking glory, but also selflessly sacrificing his own safety for the safety of others.