One can analyze "Sir Patrick Spens" in the terms you ask about because it is a narrative poem--it tells a story. This, of course, means that, rhetorically, it is much like a short story.
I'll guide you in your analysis by asking some questions and making a point or two.
First, to get at the center of the poem, if Spens and his men know about the danger of going on the sea this time of year, wouldn't the elderly advisor to the king also know about that danger? What does this suggest is going on, politically, in the ballad? Who is really responsible, then, for Spens' eventual death? Isn't it implied that this is intentional?
Ballads generally do not deal in character thoughts. They center on dialogue and action. Ballads are usually straightforward in their presentation of what characters say and do. Since the speaker is not a character in the narrative, since the second person you is not used, and since character thoughts are not revealed, what point of view is being used?
Concerning plot, what are the main events in the poem, and how do they allow other elements in the poem to work? For instance, the ship wrecks and Spens and his crew die. This is not directly stated, however--there isn't time or space in a ballad to do justice to the destruction of a ship and crew. Instead, understatement is used--their hats float on the water. The reader gets the understated, yet powerful, message.
What do the actions of the elderly advisor, the king, and Spens reveal about their characters? How does power work in the poem? How are the characters, therefore, characterized?
Finally, in general, the poem is structured as a ballad: four-line stanzas with the lines alternating between eight and six syllables, and a rhyme scheme in each stanza of abcb. It begins in medias res, in the middle of things. The reader doesn't know what the mission is or why it's necessary or what the situation is. The narrative begins as close to the death of Spens as possible--only what is necessary for the reader to understand the political intrigue going on is included.
Incidentally, the poem also uses situational irony. Spens is in a no-win situation that he cannot control. This is his world and his existence, and by extension, our world and our existence. This is what power can do.