This sonnet by Keats would give heartburn to a new critic because the essence of New Criticism is to critique a work on its own literary merits, apart from any historical context or biographical influence of the author. Keats wrote this sonnet to express the joy he felt when he read Chapman's translation of Homer's The Iliad. From a "New Critical Approach" then, you must focus primarily on the craft of poetry itself. Luckily, there is a lot there.
Focus on the imagery of the sonnet, the metaphors and similes. They are beautiful. The overall metaphor is of a traveler who has experienced many "realms of gold" (metaphor for many lands). When he reads the translation of the Iliad by Chapman, he feels like a scientist who has discovered a new planet:
Then felt I like some watcher of the skies
When a new planet swims into his ken
There is a simile (like some watcher of the skies) and personification (a new planet swims) in these lines.
You can take it from there! Look for other poetic devices that enhance the imagery and you will be a New Critic!