The narrative voice of "The Weary Blues" is in the first person. The narrator describes the events s/he experiences, but does not give us any detailed self-description.
There are a few possible reasons why we do not get a self-description of the narrator.
One might be that we should assume that the narrator is, in fact, the poet himself. This is not, however, a supportable assumption, as the narrative voice a poet crafts, however it may exploit autobiographical details, is still shaped by literary exigencies. Thus it is better to think of a narrative voice as a quasi-character in this sort of poem.
Another reason for the lack of narrative self-description is to maintain the focus of the poem on the piano player rather than on the narrator.
The narrator is known to us only as a person who listens to the piano player (and refers to the piano player in the third person -- which would be rather odd if they were the same person). The narrator does have privileged information about what happens after the performance. Consider as you analyze the poem that most members of the audience would not be likely to know:
The singer stopped playing and went to bed
While the Weary Blues echoed through his head.
He slept like a rock or a man that's dead.