I know what each of the characters mean, but I'm confuse on how to put it into a sentence.
我 - I 牛 - ox 起来 - up
“我牛起来” means “I’m gonna be better/successful/cool” In this sentence, “牛” is short for “牛逼” which is meaning of “super cool”. Why does “牛逼” mean “super cool”? You could find the answer in this video https://youtu.be/teQPusK7YFU
I think there are three possible ways to look at this.
Word play: Note how 牛起來[来] and 扭起來 (lit. to shake it up) were used interchangeably in the song, both sharing the same pronunciation here in Mandarin (niú). It is quite possible that 牛起來 is a word play on 扭起來, especially given the context that this was a song written to commemorate Chunwan this year. Together they give the song the impression of lightheartedness and vivacity.
Flexibility in parts of speech (詞類[词类]活用): If the above provides the motivation of the word play, this provides its grammatical justification. Put simply, it is the phenomenon of using a word in a certain part of speech that is different from its inherent one. This is seldom observed (but still possible) in modern Chinese, more so in classical Chinese and idioms. For example: the idiom 門[门]可羅[罗]雀 (lit. birds can be netted in front of the door), which is used to describe a state of desertion, has the character 羅 in it. It is originally a noun meaning 'net; web', but in this case it is used as if it were a verb, meaning 'to catch; to net'. Similarly, in 牛起來, the lyricist used 牛 as if it were a verb. Conservatively, you may interpret it as 'to bear the qualities of an ox; to be like an ox'. Is it to encourage listeners to shake their bodies like an ox would to its tail? Or is it to encourage them to be digiligent like an ox in the forthcoming year?* That frankly is open to interpretation.
Slang: I am not inclined to adopt this explanation. 牛 is often an abbreviation of this, which means 'awesome' in informal spoken Chinese. If this were the case, it would be using the adjective as if it were a verb in 牛起來.
*The V+起來 construction, originally meaning 'to start V', can sometimes be used by the speaker to encourage the listener to start that action.