The Anglo-Saxon poem, "The Wanderer," is an elegy. An elegy by definition is mournful, mournful either of death or of the loss of a way of life.
The tone, the speaker's attitude toward his subject--the loss of a way of life--is mournful. If you have to choose from the list you give in your question, choose resigned or bitter, but neither one really fits. Mournful is more accurate.
Not that the speaker isn't somewhat bitter, and not that he isn't somewhat resigned to his fate. You can argue either one of those. But at its center, the tone is mournful.
For instance, lines 99-102 show bitterness and resignation:
"...fate's decrees transform the world.
Here wealth is fleeting, friends are fleeting,
Man is fleeting, maid is fleeting;
All the foundation of earth shall fall!"
The speaker is both resigned and bitter.
Yet, taken as a whole (without the first and last stanzas, which are interpolations), the poem is mournful. The speaker dreams of his friends, then despairs when he wakes to find he's dreaming. He is "lonely-hearted" and spends much time reflecting on his past and pondering. He writes that "proud warriors vanish," and structures he once knew "stand empty of life." He muses on "moldering ruins." He remembers where his master fell while defending his walls.
The speaker mourns, but is still thinking and pondering and creating. There is life in the poem, and art. He remembers, and his memories are vivid and possess force. The emphasis is on the past that he misses, rather than on blaming someone or something, or on resignation. This speaker is still alive and kicking, as they say, though he is in mourning.